They have the same degrees, but the exponents of the degrees are not the same, 2² * 2³, then the result will be a base of the degree with the same identical base of the terms of the product of degrees, raised to an exponent equal to the sum of the exponents of all multiplied degrees.

2² * 2³ = 2²⁺³ = 2⁵ = 32

If the terms of a product of powers have different bases of powers, and the exponents are the same, for example, 2³ * 5³, then the result will be the product of the bases of these powers raised to an exponent equal to this same exponent.

2³ * 5³ = (2*5)³ = 10³ = 1000

If the powers being multiplied are equal to each other, for example, 5³ * 5³, then the result will be a power with a base equal to these identical bases of powers, raised to an exponent equal to the exponent of the powers, multiplied by the number of these identical powers.

5³ * 5³ = (5³)² = 5³*² = 5⁶ = 15625

Or another example with the same result:

5² * 5² * 5² = (5²)³ = 5²*³ = 5⁶ = 15625

Sources:

• What is a degree with a natural exponent?
• product of powers

Mathematical operations with powers can be performed only when the bases of the exponents are the same, and when there are multiplication or division signs between them. The base of an exponent is the number that is raised to a power.

Instructions

If the numbers are divisible by each other (cm 1), then y (in this example, this is the number 3) appears as a power, which is formed by subtracting the exponents. Moreover, this action is carried out directly: the second is subtracted from the first indicator. Example 1. Let us introduce: (a)b, where in brackets – a is the base, outside brackets – in – the exponent. (6)5: (6)3 = (6)5-3 = (6) 2 = 6*6 = 36. If the answer produces a number to a negative power, then such a number is converted to common fraction, in the numerator of which there is a unit, and in the denominator the base with the exponent obtained from the difference, only in positive form (with a plus sign). Example 2. (2) 4: (2)6 = (2) 4-6 = (2) -2 = 1/(2)2 = ¼. The division of powers can be written in a different form, through the fraction sign, and not as indicated in this step through the “:” sign. This does not change the principle of the solution, everything is done exactly the same, only the entry will be made with a horizontal (or oblique) fraction sign instead of a colon. Example 3. (2) 4 / (2)6 = (2) 4-6 = (2 ) -2 = 1/(2)2 = ¼.

When multiplying identical bases that have degrees, the degrees are added. Example 4. (5) 2* (5)3 = (5)2+3 = (5)5 = 3125. If the exponents have different signs, then their addition is carried out according to mathematical laws. Example 5. (2)1* (2)-3 = (2) 1+(-3) = (2) -2 = 1/(2)2 = ¼.

If the bases of exponents differ, then most likely they can be brought to the same form by mathematical transformation. Example 6. Suppose we need to find the value of the expression: (4)2: (2)3. Knowing that the number four can be represented as two squared, this example is solved as follows: (4)2: (2)3 = (2*2)2: (2)3. Next, when raising a number to a power. Already having a degree, the degree indices are multiplied by each other: ((2)2)2: (2)3 = (2)4: (2)3 = (2) 4-3 = (2)1 = 2.

Helpful advice

Remember, if a given base seems different from the second base, look for a mathematical solution. Different numbers are not just given. Unless the typesetter made a typo in the textbook.

The power format of writing a number is a shortened form of writing the operation of multiplying a base by itself. With a number presented in this form, you can perform the same operations as with any other numbers, including raising them to a power. For example, you can raise the square of a number to an arbitrary power and obtaining the result at the current level of technological development will not pose any difficulty.

You will need

• Internet access or Windows calculator.

Instructions

To raise a square to a power, use general rule raising to a power that already has power exponent. With this operation, the indicators are multiplied, but the base remains the same. If the base is designated as x, and the original and additional indicators are designated as a and b, write this rule in general view you can do this: (xᵃ)ᵇ=xᵃᵇ.

How to multiply powers? Which powers can be multiplied and which cannot? How to multiply a number by a power?

In algebra, you can find a product of powers in two cases:

1) if the degrees have the same bases;

2) if the degrees have the same indicators.

When multiplying powers with the same bases, the base must be left the same, and the exponents must be added:

When multiplying degrees with the same indicators, the overall indicator can be taken out of brackets:

Let's look at how to multiply powers using specific examples.

The unit is not written in the exponent, but when multiplying powers, they take into account:

When multiplying, there can be any number of powers. It should be remembered that you don’t have to write the multiplication sign before the letter:

In expressions, exponentiation is done first.

If you need to multiply a number by a power, you should first perform the exponentiation, and only then the multiplication:

www.algebraclass.ru

Addition, subtraction, multiplication, and division of powers

Addition and subtraction of powers

It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds equal powers of identical variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is equal to 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3.

It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 — 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Multiplying powers

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.

Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the multiplication result, which is equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n;

And a m is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents of the powers.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x – 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y -n .y -m = y -n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers equal to the sum or the difference of their squares.

If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.

So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.

Division of degrees

Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.

Thus, a 3 b 2 divided by b 2 is equal to a 3.

Writing a 5 divided by a 3 looks like $\frac $. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing degrees with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1. That is, $\frac = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.

Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also true for numbers with negative values ​​of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$

It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Decrease the exponents by $\frac $ Answer: $\frac $.

2. Decrease exponents by $\frac$. Answer: $\frac$ or 2x.

3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and lead to common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

Properties of degree

We remind you that in this lesson we will understand properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.

A degree with a natural indicator has several important properties, which allow you to simplify calculations in examples with powers.

Property No. 1
Product of powers

When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.

a m · a n = a m + n, where “a” is any number, and “m”, “n” are any integers.

This property of powers also applies to the product of three or more powers.

Simplify the expression.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15

Present it as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17

Present it as a degree.
(0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Please note that in specified property we were talking only about multiplying powers with the same bases. It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243

Property No. 2
Partial degrees

When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2

Calculate.

11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4

Answer: t = 3 4 = 81

Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

Example. Find the value of an expression using the properties of exponents.

2 11 − 5 = 2 6 = 64

Please note that in Property 2 we were only talking about dividing powers with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4

Property No. 3
Raising a degree to a power

When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.

Please note that property No. 4, like other properties of degrees, is also applied in reverse order.

(a n · b n)= (a · b) n

That is, to multiply powers with the same exponents, you can multiply the bases, but leave the exponent unchanged.

Example. Calculate.
2 4 5 4 = (2 5) 4 = 10 4 = 10,000

Example. Calculate.
0.5 16 2 16 = (0.5 2) 16 = 1

In more complex examples, there may be cases when multiplication and division must be performed on powers with different bases and different indicators. In this case, we advise you to do the following.

For example, 4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216

An example of raising a decimal to a power.

4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = 4

Properties 5
Power of a quotient (fraction)

To raise a quotient to a power, you can raise the dividend and the divisor separately to this power, and divide the first result by the second.

(a: b) n = a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n - any natural number.

Example. Present the expression as a quotient of powers.
(5: 3) 12 = 5 12: 3 12

We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

Powers and roots

Operations with powers and roots. Degree with negative ,

zero and fractional indicator. About expressions that have no meaning.

Operations with degrees.

1. When multiplying powers with the same base, their exponents are added:

a m · a n = a m + n .

2. When dividing degrees with the same base, their exponents are deducted .

3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.

4. The degree of a ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):

(a/b) n = a n / b n .

5. When raising a power to a power, their exponents are multiplied:

All the above formulas are read and executed in both directions from left to right and vice versa.

EXAMPLE (2 3 5 / 15)² = 2² · 3² · 5² / 15² = 900 / 225 = 4 .

Operations with roots. In all the formulas below, the symbol means arithmetic root(the radical expression is positive).

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. Root of attitude equal to the ratio roots of dividend and divisor:

3. When raising a root to a power, it is enough to raise to this power radical number:

4. If you increase the degree of the root by m times and at the same time raise the radical number to the mth power, then the value of the root will not change:

5. If you reduce the degree of the root by m times and simultaneously extract the mth root of the radical number, then the value of the root will not change:

Expanding the concept of degree. So far we have considered degrees only with natural exponents; but operations with powers and roots can also lead to negative, zero And fractional indicators. All these exponents require additional definition.

A degree with a negative exponent. The power of a certain number with a negative (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the negative exponent:

Now the formula a m : a n = a m - n can be used not only for m, more than n, but also with m, less than n .

EXAMPLE a 4: a 7 = a 4 — 7 = a — 3 .

If we want the formula a m : a n = a m — n was fair when m = n, we need a definition of degree zero.

A degree with a zero index. The power of any non-zero number with exponent zero is 1.

EXAMPLES. 2 0 = 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.

Degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the nth root of the mth power of this number a:

About expressions that have no meaning. There are several such expressions.

Where a ≠ 0 , does not exist.

In fact, if we assume that x is a certain number, then in accordance with the definition of the division operation we have: a = 0· x, i.e. a= 0, which contradicts the condition: a ≠ 0

— any number.

In fact, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 = 0 · x. But this equality occurs when any number x, which was what needed to be proven.

0 0 — any number.

Solution. Let's consider three main cases:

1) x = 0 – this value does not satisfy this equation

2) when x> 0 we get: x/x= 1, i.e. 1 = 1, which means

What x– any number; but taking into account that in

in our case x> 0, the answer is x > 0 ;

Rules for multiplying powers with different bases

DEGREE WITH RATIONAL INDICATOR,

POWER FUNCTION IV

§ 69. Multiplication and division of powers with the same bases

Theorem 1. To multiply powers with the same bases, it is enough to add the exponents and leave the base the same, that is

Proof. By definition of degree

2 2 2 3 = 2 5 = 32; (-3) (-3) 3 = (-3) 4 = 81.

We looked at the product of two powers. In fact, the proven property is true for any number of powers with the same bases.

Theorem 2. To divide powers with the same bases, when the index of the dividend is greater than the index of the divisor, it is enough to subtract the index of the divisor from the index of the dividend, and leave the base the same, that is at t > p

(a =/= 0)

Proof. Recall that the quotient of dividing one number by another is the number that, when multiplied by the divisor, gives the dividend. Therefore, prove the formula where a =/= 0, it's the same as proving the formula

If t > p , then the number t - p will be natural; therefore, by Theorem 1

Theorem 2 is proven.

It should be noted that the formula

we have proved it only under the assumption that t > p . Therefore, from what has been proven, it is not yet possible to draw, for example, the following conclusions:

In addition, we have not yet considered degrees with negative exponents and we do not yet know what meaning can be given to expression 3 - 2 .

Theorem 3. To raise a degree to a power, it is enough to multiply the exponents, leaving the base of the degree the same, that is

Proof. Using the definition of degree and Theorem 1 of this section, we obtain:

Q.E.D.

For example, (2 3) 2 = 2 6 = 64;

518 (Oral) Determine X from the equations:

1) 2 2 2 2 3 2 4 2 5 2 6 = 2 x ; 3) 4 2 4 4 4 6 4 8 4 10 = 2 x ;

2) 3 3 3 3 5 3 7 3 9 = 3 x ; 4) 1 / 5 1 / 25 1 / 125 1 / 625 = 1 / 5 x .

519. (Set no.) Simplify:

520. (Set no.) Simplify:

521. Present these expressions in the form of degrees with the same bases:

1) 32 and 64; 3) 8 5 and 16 3; 5) 4 100 and 32 50;

2) -1000 and 100; 4) -27 and -243; 6) 81 75 8 200 and 3 600 4 150.

Basic properties of degrees

"Properties of degrees" is a fairly popular query in search engines, which shows great interest in the properties of the degree. We have collected for you all the properties of a degree (properties of a degree with a natural exponent, properties of a degree with rational indicator, properties of a degree with an integer exponent) in one place. You can download a short version of the cheat sheet "Properties of degrees" in .pdf format so that, if necessary, you can easily remember them or familiarize yourself with them properties of degrees directly on the site. In details properties of powers with examples discussed below.

Download the cheat sheet "Properties of degrees" (format.pdf)

Properties of degrees (briefly)

a 0=1 if a≠0

a 1=a

(−a)n=an, If n- even

(−a)n=−an, If n- odd

(a⋅b)n=an⋅bn

(ab)n=anbn

a−n=1an

(ab)−n=(ba)n

an⋅am=an+m

anam=an−m

(an)m=an⋅m

Properties of powers (with examples)

1st degree property Any number other than zero to the zero power is equal to one. a 0=1 if a≠0 For example: 1120=1, (−4)0=1, (0,15)0=1

2nd degree property Any number to the first power is equal to the number itself. a 1=a For example: 231=23, (−9,3)1=−9,3

3rd degree property Any number to an even power is positive. an=an, If n- even (divisible by 2) integer (− a)n=an, If n- even (divisible by 2) integer For example: 24=16, (−3)2=32=9, (−1)10=110=1

4th degree property Any number to an odd power retains its sign. an=an, If n- odd (not divisible by 2) integer (− a)n=−an, If n- odd (not divisible by 2) integer For example: 53=125, (−3)3=33=27, (−1)11=−111=−1

5th degree property Product of numbers raised oh to a power, can be represented as the product of numbers raised s V this degree (and vice versa). ( a⋅b)n=an⋅bn, wherein a, b, n For example: (2,1⋅0,3)4,5=2,14,5⋅0,34,5

6th degree property The quotient (division) of numbers raised oh to a power, can be represented as the quotient of numbers raised s V this degree (and vice versa). ( ab)n=anbn, wherein a, b, n- any valid (not necessarily integer) numbers For example: (1,75)0,1=(1,7)0,150,1

7th degree property Any number to a negative power is equal to its reciprocal number to that power. (The reciprocal is the number by which the given number must be multiplied to get one.) a−n=1an, wherein a And n- any valid (not necessarily integer) numbers For example: 7−2=172=149

8th degree property Any fraction to a negative power is equal to the reciprocal fraction to that power. ( ab)−n=(ba)n, wherein a, b, n- any valid (not necessarily integer) numbers For example: (23)−2=(32)2, (14)−3=(41)3=43=64

9th degree property When multiplying powers with the same base, the exponents are added, but the base remains the same. an⋅am=an+m, wherein a, n, m- any valid (not necessarily integer) numbers For example: 23⋅25=23+5=28, note that this property of the degree is preserved for negative values ​​of the degrees 3−2⋅36=3−2+6=34, 47⋅4−3=47+(−3)= 47−3=44

10th degree property When dividing powers with the same base, the exponents are subtracted, but the base remains the same. anam=an−m, wherein a, n, m- any valid (not necessarily integer) numbers For example:(1,4)2(1,4)3=1.42+3=1.45, note how this power property applies to negative powers3−236=3−2−6=3−8, 474− 3=47−(−3)=47+3=410

11th degree property When raising a power to a power, the powers are multiplied. ( an)m=an⋅m For example: (23)2=23⋅2=26=64

Table of powers up to 10

Few people manage to remember the entire table of degrees, and who needs it when it is so easy to find? Our power table includes both the popular tables of squares and cubes (from 1 to 10), as well as tables of other powers that are less common. The columns of the table of powers indicate the bases of the degree (the number that needs to be raised to a power), the rows indicate the exponents (the power to which the number needs to be raised), and at the intersection of the desired column and the desired row is the result of raising the desired number to a given power. There are several types of problems that can be solved using power tables. The immediate task is to calculate n th power of a number. The inverse problem, which can also be solved using a table of powers, may sound like this: “to what power should the number be raised? a to get the number b ?" or "What number to the power n gives a number b ?".

Table of powers up to 10

	1 n	2 n	3 n	4 n	5 n	6 n	7 n	8 n	9 n	10 n

How to use the degree table

Let's look at a few examples of using the power table.

Example 1. What number results from raising the number 6 to the 8th power? In the table of degrees we look for column 6 n, since according to the conditions of the problem the number 6 is raised to a power. Then in the table of powers we look for line 8, since the given number must be raised to the power of 8. At the intersection we look at the answer: 1679616.

Example 2. To what power must the number 9 be raised to get 729? In the table of degrees we look for column 9 n and we go down it to the number 729 (the third line of our table of degrees). The line number is the required degree, that is, the answer: 3.

Example 3. What number must be raised to the power of 7 to get 2187? In the table of degrees we look for line 7, then move along it to the right to the number 2187. From the found number we go up and find out that the heading of this column is 3 n, which means the answer is: 3.

Example 4. To what power must the number 2 be raised to get 63? In the table of degrees we find column 2 n and we go down it until we meet 63... But this will not happen. We will never see the number 63 in this column or in any other column of the table of powers, which means that no integer from 1 to 10 gives the number 63 when raised to an integer power from 1 to 10. Thus, there is no answer .

Why are degrees needed?

Where will you need them?

Why should you take the time to study them?

To find out EVERYTHING ABOUT DEGREES, read this article.

And, of course, knowledge of degrees will bring you closer to successfully passing the Unified State Exam.

And to admission to the university of your dreams!

Let's go... (Let's go!)

FIRST LEVEL

Exponentiation is a mathematical operation just like addition, subtraction, multiplication or division.

Now I will explain everything in human language in very simple examples. Be careful. The examples are elementary, but explain important things.

Let's start with addition.

There is nothing to explain here. You already know everything: there are eight of us. Everyone has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written differently: . Mathematicians are cunning and lazy people. They first notice some patterns, and then figure out a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table. Of course, you can do everything slower, more difficult and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful one:

What other clever counting tricks have lazy mathematicians come up with? Right - raising a number to a power.

Raising a number to a power

If you need to multiply a number by itself five times, then mathematicians say that you need to raise that number to the fifth power. For example, . Mathematicians remember that two to the fifth power is... And they solve such problems in their heads - faster, easier and without mistakes.

All you need to do is remember what is highlighted in color in the table of powers of numbers. Believe me, this will make your life a lot easier.

By the way, why is it called the second degree? square numbers, and the third - cube? What does it mean? Very good question. Now you will have both squares and cubes.

Real life example #1

Let's start with the square or the second power of the number.

Imagine a square pool measuring one meter by one meter. The pool is at your dacha. It's hot and I really want to swim. But... the pool has no bottom! You need to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the bottom area of ​​the pool.

You can simply calculate by pointing your finger that the bottom of the pool consists of meter by meter cubes. If you have tiles one meter by one meter, you will need pieces. It's easy... But where have you seen such tiles? The tile will most likely be cm by cm. And then you will be tortured by “counting with your finger.” Then you have to multiply. So, on one side of the bottom of the pool we will fit tiles (pieces) and on the other, too, tiles. Multiply by and you get tiles ().

Did you notice that to determine the area of ​​the pool bottom we multiplied the same number by itself? What does it mean? Since we are multiplying the same number, we can use the “exponentiation” technique. (Of course, when you have only two numbers, you still need to multiply them or raise them to a power. But if you have a lot of them, then raising them to a power is much easier and there are also fewer errors in calculations. For the Unified State Exam, this is very important).
So, thirty to the second power will be (). Or we can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. And vice versa, if you see a square, it is ALWAYS the second power of some number. A square is an image of the second power of a number.

Real life example #2

Here's a task for you: count how many squares there are on the chessboard using the square of the number... On one side of the cells and on the other too. To calculate their number, you need to multiply eight by eight or... if you notice that a chessboard is a square with a side, then you can square eight. You will get cells. () So?

Real life example #3

Now the cube or the third power of a number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Unexpected, right?) Draw a pool: the bottom is a meter in size and a meter deep, and try to count how many cubes measuring a meter by a meter will fit into your pool.

Just point your finger and count! One, two, three, four...twenty-two, twenty-three...How many did you get? Not lost? Is it difficult to count with your finger? So that! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes... Easier, right?

Now imagine how lazy and cunning mathematicians are if they simplified this too. We reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself... What does this mean? This means you can take advantage of the degree. So, what you once counted with your finger, they do in one action: three cubed is equal. It is written like this: .

All that remains is remember the table of degrees. Unless, of course, you are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can continue to count with your finger.

Well, to finally convince you that degrees were invented by quitters and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Real life example #4

You have a million rubles. At the beginning of each year, for every million you make, you make another million. That is, every million you have doubles at the beginning of each year. How much money will you have in years? If you are sitting now and “counting with your finger,” then you are a very hardworking person and... stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two multiplied by two... in the second year - what happened, by two more, in the third year... Stop! You noticed that the number is multiplied by itself times. So two to the fifth power is a million! Now imagine that you have a competition and the one who can count the fastest will get these millions... It’s worth remembering the powers of numbers, don’t you think?

Real life example #5

You have a million. At the beginning of each year, for every million you make, you earn two more. Great isn't it? Every million is tripled. How much money will you have in a year? Let's count. The first year - multiply by, then the result by another... It’s already boring, because you already understood everything: three is multiplied by itself times. So to the fourth power it is equal to a million. You just have to remember that three to the fourth power is or.

Now you know that by raising a number to a power you will make your life a lot easier. Let's take a further look at what you can do with degrees and what you need to know about them.

Terms and concepts... so as not to get confused

So, first, let's define the concepts. What do you think, what is an exponent? It's very simple - it's the number that is "at the top" of the power of the number. Not scientific, but clear and easy to remember...

Well, at the same time, what such a degree basis? Even simpler - this is the number that is located below, at the base.

Here's a drawing for good measure.

Well, in general terms, in order to generalize and remember better... A degree with a base “ ” and an exponent “ ” is read as “to the degree” and is written as follows:

Power of a number with natural exponent

You probably already guessed: because the exponent is a natural number. Yes, but what is it natural number? Elementary! Natural numbers are those numbers that are used in counting when listing objects: one, two, three... When we count objects, we do not say: “minus five,” “minus six,” “minus seven.” We also do not say: “one third”, or “zero point five”. These are not natural numbers. What numbers do you think these are?

Numbers like “minus five”, “minus six”, “minus seven” refer to whole numbers. In general, integers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and number. Zero is easy to understand - it is when there is nothing. What do negative (“minus”) numbers mean? But they were invented primarily to indicate debts: if you have a balance on your phone in rubles, this means that you owe the operator rubles.

All fractions are rational numbers. How did they arise, do you think? Very simple. Several thousand years ago, our ancestors discovered that they lacked natural numbers to measure length, weight, area, etc. And they came up with rational numbers... Interesting, isn't it?

There are also irrational numbers. What are these numbers? In short, endless decimal. For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Summary:

Let us define the concept of a degree whose exponent is a natural number (i.e., integer and positive).

1. Any number to the first power is equal to itself:
2. To square a number means to multiply it by itself:
3. To cube a number means to multiply it by itself three times:

Definition. Raising a number to a natural power means multiplying the number by itself times:
.

Properties of degrees

Where did these properties come from? I will show you now.

Let's see: what is it And ?

A-priory:

How many multipliers are there in total?

It’s very simple: we added multipliers to the factors, and the result is multipliers.

But by definition, this is a power of a number with an exponent, that is: , which is what needed to be proven.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule Necessarily there must be the same reasons!
Therefore, we combine the powers with the base, but it remains a separate factor:

only for the product of powers!

Under no circumstances can you write that.

2. that's it th power of a number

Just as with the previous property, let us turn to the definition of degree:

It turns out that the expression is multiplied by itself times, that is, according to the definition, this is the th power of the number:

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total:

Let's remember the abbreviated multiplication formulas: how many times did we want to write?

But this is not true, after all.

Power with negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the basis?

In powers of natural indicator the basis may be any number. Indeed, we can multiply any numbers by each other, be they positive, negative, or even.

Let's think about which signs ("" or "") will have degrees of positive and negative numbers?

For example, is the number positive or negative? A? ? With the first one, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for minus gives a plus.” That is, or. But if we multiply by, it works.

Determine for yourself what sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Did you manage?

Here are the answers: In the first four examples, I hope everything is clear? We simply look at the base and exponent and apply the appropriate rule.

In example 5) everything is also not as scary as it seems: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive.

Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple!

6 examples to practice

Analysis of the solution 6 examples

Whole we call the natural numbers, their opposites (that is, taken with the " " sign) and the number.

positive integer, and it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at new cases. Let's start with an indicator equal to.

Any number to the zero power is equal to one:

As always, let us ask ourselves: why is this so?

Let's consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply by so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we cannot not only divide by zero, but also raise it to the zero power.

Let's move on. In addition to natural numbers and numbers, integers also include negative numbers. To understand what a negative power is, let’s do as last time: multiply some normal number by the same number to a negative power:

From here it’s easy to express what you’re looking for:

Now let’s extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number with a negative power is the reciprocal of the same number with a positive power. But at the same time The base cannot be null:(because you can’t divide by).

Let's summarize:

Tasks for independent solution:

Well, as usual, examples for independent solutions:

Analysis of problems for independent solution:

I know, I know, the numbers are scary, but on the Unified State Exam you have to be prepared for anything! Solve these examples or analyze their solutions if you couldn’t solve them and you will learn to cope with them easily in the exam!

Let's continue to expand the range of numbers “suitable” as an exponent.

Now let's consider rational numbers. What numbers are called rational?

Answer: everything that can be represented as a fraction, where and are integers, and.

To understand what it is "fractional degree", consider the fraction:

Let's raise both sides of the equation to a power:

Now let's remember the rule about "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse operation of raising to a power: .

It turns out that. Obviously this special case can be expanded: .

Now we add the numerator: what is it? The answer is easy to obtain using the power-to-power rule:

But can the base be any number? After all, the root cannot be extracted from all numbers.

None!

Remember the rule: any number raised to even degree- the number is positive. That is, it is impossible to extract even roots from negative numbers!

This means that such numbers cannot be raised to fractional power with an even denominator, that is, the expression does not make sense.

What about the expression?

But here a problem arises.

The number can be represented in the form of other, reducible fractions, for example, or.

And it turns out that it exists, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write it down. But if we write down the indicator differently, we will again get into trouble: (that is, we got a completely different result!).

To avoid such paradoxes, we consider only positive base exponent with fractional exponent.

So if:

• - natural number;
• - integer;

Examples:

Rational exponents are very useful for transforming expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

Well, now comes the hardest part. Now we'll figure it out degree with irrational exponent.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception

After all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms.

For example, a degree with a natural exponent is a number multiplied by itself several times;

...number to the zeroth power- this is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number;

...negative integer degree- it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

By the way, in science a degree with a complex exponent is often used, that is, the exponent is not even a real number.

But at school we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the usual rule for raising a power to a power:

ADVANCED LEVEL

Determination of degree

A degree is an expression of the form: , where:

• — degree base;
• - exponent.

Degree with natural indicator (n = 1, 2, 3,...)

Raising a number to the natural power n means multiplying the number by itself times:

Degree with an integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

Construction to the zero degree:

The expression is indefinite, because, on the one hand, to any degree is this, and on the other hand, any number to the th degree is this.

If the exponent is negative integer number:

(because you can’t divide by).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Power with rational exponent

• - natural number;
• - integer;

Examples:

Properties of degrees

To make it easier to solve problems, let’s try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression we get the following product:

But by definition it is a power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule Necessarily there must be the same reasons. Therefore, we combine the powers with the base, but it remains a separate factor:

Another important note: this rule - only for product of powers!

Under no circumstances can you write that.

Just as with the previous property, let us turn to the definition of degree:

Let's regroup this work like this:

It turns out that the expression is multiplied by itself times, that is, according to the definition, this is the th power of the number:

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total: !

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

Power with a negative base.

Up to this point we have only discussed what it should be like index degrees. But what should be the basis? In powers of natural indicator the basis may be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have degrees of positive and negative numbers?

For example, is the number positive or negative? A? ?

With the first one, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for minus gives a plus.” That is, or. But if we multiply by (), we get - .

And so on ad infinitum: with each subsequent multiplication the sign will change. We can formulate the following simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any degree is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We simply look at the base and exponent and apply the appropriate rule.

In example 5) everything is also not as scary as it seems: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive. Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If we remember that, it becomes clear that, which means the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and divide them by each other, divide them into pairs and get:

Before we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

Let's go back to the example:

And again the formula:

So now the last rule:

How will we prove it? Of course, as usual: let’s expand on the concept of degree and simplify it:

Well, now let's open the brackets. How many letters are there in total? times by multipliers - what does this remind you of? This is nothing more than a definition of an operation multiplication: There were only multipliers there. That is, this, by definition, is a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about degrees for the average level, we will analyze the degree with an irrational exponent. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational numbers).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms. For example, a degree with a natural exponent is a number multiplied by itself several times; a number to the zero power is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number; a degree with an integer negative exponent - it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). It is rather a purely mathematical object that mathematicians created to extend the concept of degree to the entire space of numbers.

By the way, in science a degree with a complex exponent is often used, that is, the exponent is not even a real number. But at school we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree called an expression of the form: , where:

Degree with an integer exponent

a degree whose exponent is a natural number (i.e., integer and positive).

Power with rational exponent

degree, the exponent of which is negative and fractional numbers.

Degree with irrational exponent

a degree whose exponent is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any degree is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE THE WORD...

How do you like the article? Write below in the comments whether you liked it or not.

Tell us about your experience using degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck on your exams!

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After the power of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.

Page navigation.

Properties of degrees with natural exponents

By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, we can obtain and justify the following properties of degree with natural exponent:

1. the main property of the degree a m ·a n =a m+n, its generalization;
2. property of quotient powers with identical bases a m:a n =a m−n ;
3. product power property (a·b) n =a n ·b n , its extension;
4. property of the quotient to the natural degree (a:b) n =a n:b n ;
5. raising a degree to a power (a m) n =a m·n, its generalization (((a n 1) n 2) …) n k =a n 1 ·n 2 ·…·n k;
6. comparison of degree with zero:
   • if a>0, then a n>0 for any natural number n;
   • if a=0, then a n =0;
   • if a<0 и показатель степени является четным числом 2·m , то a 2·m >0 if a<0 и показатель степени есть нечетное число 2·m−1 , то a 2·m−1 <0 ;
7. if a and b are positive numbers and a
8. if m and n are natural numbers such that m>n , then at 0 0 the inequality a m >a n is true.

Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying expressions often used in the form a m+n =a m ·a n .

Now let's look at each of them in detail.

Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.

Let us prove the main property of the degree. By the definition of a power with a natural exponent, the product of powers with the same bases of the form a m ·a n can be written as a product. Due to the properties of multiplication, the resulting expression can be written as , and this product is a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.

Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values ​​of the expressions 2 2 · 2 3 and 2 5 . Performing exponentiation, we have 2 2 ·2 3 =(2·2)·(2·2·2)=4·8=32 and 2 5 =2·2·2·2·2=32, since equal values ​​are obtained, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.

The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1, n 2, …, n k the following equality is true: a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k.

For example, (2,1) 3 ·(2,1) 3 ·(2,1) 4 ·(2,1) 7 = (2,1) 3+3+4+7 =(2,1) 17 .

We can move on to the next property of powers with a natural exponent – property of quotient powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n, the equality a m:a n =a m−n is true.

Before presenting the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that we cannot divide by zero. The condition m>n is introduced so that we do not go beyond the natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n) or a negative number (which happens for m

Proof. The main property of a fraction allows us to write the equality a m−n ·a n =a (m−n)+n =a m. From the resulting equality a m−n ·a n =a m and it follows that a m−n is a quotient of the powers a m and a n . This proves the property of quotient powers with identical bases.

Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the equality π 5:π 2 =π 5−3 =π 3 corresponds to the considered property of the degree.

Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n .

Indeed, by the definition of a degree with a natural exponent we have . Based on the properties of multiplication, the last product can be rewritten as , which is equal to a n · b n .

Here's an example: .

This property extends to the power of the product of three or more factors. That is, the property of natural degree n of the product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n.

For clarity, we will show this property with an example. For the product of three factors to the power of 7 we have .

The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of powers a n and b n, that is, (a:b) n =a n:b n.

The proof can be carried out using the previous property. So (a:b) n b n =((a:b) b) n =a n, and from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of a n divided by b n .

Let's write this property using specific numbers as an example: .

Now let's voice it property of raising a power to a power: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of the number a with exponent m·n, that is, (a m) n =a m·n.

For example, (5 2) 3 =5 2·3 =5 6.

The proof of the power-to-degree property is the following chain of equalities: .

The property considered can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality . For greater clarity, here is an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10 .

It remains to dwell on the properties of comparing degrees with a natural exponent.

Let's start by proving the property of comparing zero and power with a natural exponent.

First, let's prove that a n >0 for any a>0.

The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a, the degree a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and .

It is quite obvious that for any natural number n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0.

Let's move on to negative bases of degree.

Let's start with the case when the exponent is an even number, let's denote it as 2·m, where m is a natural number. Then . For each of the products of the form a·a is equal to the product of the moduli of the numbers a and a, which means it is a positive number. Therefore, the product will also be positive and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .

Finally, when the base a is a negative number and the exponent is an odd number 2 m−1, then . All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining a negative number a results in a negative number. Due to this property (−5) 3<0 , (−0,003) 17 <0 и .

Let's move on to the property of comparing powers with the same natural exponents, which has the following formulation: of two powers with the same natural exponents, n is less than the one whose base is smaller, and greater is the one whose base is larger. Let's prove it.

Inequality a n properties of inequalities a provable inequality of the form a n is also true (2.2) 7 and .

It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of two powers with natural exponents and identical positive bases less than one, the one whose exponent is smaller is greater; and of two powers with natural exponents and identical bases greater than one, the one whose exponent is greater is greater. Let us proceed to the proof of this property.

Let us prove that for m>n and 0 0 due to the initial condition m>n, which means that at 0

It remains to prove the second part of the property. Let us prove that for m>n and a>1 a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1 the degree a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.

Properties of powers with integer exponents

Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.

We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.

So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true: properties of powers with integer exponents:

1. a m ·a n =a m+n ;
2. a m:a n =a m−n ;
3. (a·b) n =a n ·b n ;
4. (a:b) n =a n:b n ;
5. (a m) n =a m·n ;
6. if n is a positive integer, a and b are positive numbers, and a b−n ;
7. if m and n are integers, and m>n , then at 0 1 the inequality a m >a n holds.

When a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.

Proving each of these properties is not difficult; to do this, it is enough to use the definitions of degrees with natural and integer exponents, as well as the properties of operations with real numbers. As an example, let us prove that the power-to-power property holds for both positive integers and non-positive integers. To do this, you need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (a p) q =a p·q, (a −p) q =a (−p)·q, (a p ) −q =a p·(−q) and (a −p) −q =a (−p)·(−q). Let's do it.

For positive p and q, the equality (a p) q =a p·q was proven in the previous paragraph. If p=0, then we have (a 0) q =1 q =1 and a 0·q =a 0 =1, whence (a 0) q =a 0·q. Similarly, if q=0, then (a p) 0 =1 and a p·0 =a 0 =1, whence (a p) 0 =a p·0. If both p=0 and q=0, then (a 0) 0 =1 0 =1 and a 0·0 =a 0 =1, whence (a 0) 0 =a 0·0.

Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then . By the property of quotients to powers we have . Since 1 p =1·1·…·1=1 and , then . The last expression, by definition, is a power of the form a −(p·q), which, due to the rules of multiplication, can be written as a (−p)·q.

Likewise .

AND .

Using the same principle, you can prove all other properties of a degree with an integer exponent, written in the form of equalities.

In the penultimate of the recorded properties, it is worth dwelling on the proof of the inequality a −n >b −n, which is valid for any negative integer −n and any positive a and bfor which the condition a is satisfied . Since by condition a 0 . The product a n · b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as the quotient of the positive numbers b n −a n and a n ·b n . Therefore, whence a −n >b −n , which is what needed to be proved.

The last property of powers with integer exponents is proved in the same way as a similar property of powers with natural exponents.

Properties of powers with rational exponents

We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers with integer exponents. Namely:

The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, and on the properties of a degree with an integer exponent. Let us provide evidence.

By definition of a power with a fractional exponent and , then . The properties of the arithmetic root allow us to write the following equalities. Further, using the property of a degree with an integer exponent, we obtain , from which, by the definition of a degree with a fractional exponent, we have , and the indicator of the degree obtained can be transformed as follows: . This completes the proof.

The second property of powers with fractional exponents is proved in an absolutely similar way:

The remaining equalities are proved using similar principles:

Let's move on to proving the next property. Let us prove that for any positive a and b, a b p . Let's write the rational number p as m/n, where m is an integer and n is a natural number. Conditions p<0 и p>0 in this case the conditions m<0 и m>0 accordingly. For m>0 and a

Similarly, for m<0 имеем a m >b m , from where, that is, and a p >b p .

It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q, p>q at 0 0 – inequality a p >a q . We can always reduce rational numbers p and q to a common denominator, even if we get ordinary fractions and , where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from. Then, by the property of comparing powers with the same bases and natural exponents at 0 1 – inequality a m 1 >a m 2 . These inequalities in the properties of the roots can be rewritten accordingly as And . And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From here we draw the final conclusion: for p>q and 0 0 – inequality a p >a q .

Properties of powers with irrational exponents

From the way a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0, b>0 and irrational numbers p and q the following are true properties of powers with irrational exponents:

1. a p ·a q =a p+q ;
2. a p:a q =a p−q ;
3. (a·b) p =a p ·b p ;
4. (a:b) p =a p:b p ;
5. (a p) q =a p·q ;
6. for any positive numbers a and b, a 0 the inequality a p b p ;
7. for irrational numbers p and q, p>q at 0 0 – inequality a p >a q .

From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.

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