have the same exponents, and the exponents are not the same, 2² * 2³, then the result will be the base of the degree with the same same base of the terms of the product of degrees raised to exponent, equal to the sum of the indicators of all multiplied degrees.
2² * 2³ = 2²⁺³ = 2⁵ = 32
If the terms of a product of exponents have different bases, and the exponents are the same, for example, 2³ * 5³ , then the result is the product of the bases of these powers raised to an exponent equal to that same exponent.
2³ * 5³ = (2*5)³ = 10³ = 1000
If the multiplied exponents are equal to each other, for example, 5³ * 5³, then the result will be a degree with a base equal to these identical bases of degrees, raised to an exponent equal to the exponent, multiplied by the number of these identical degrees.
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 indicator
- product of powers
Mathematical operations with powers can only be performed when the bases of the exponents are the same, and when there are multiplication or division signs between them. The base of the exponent is the number that is raised to the power.
Instruction
If the numbers are divided by each other (see 1), then y (in this example, this is the number 3) has a degree, 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's introduce: (а) в, where in brackets - a - the base, outside brackets - in - the exponent. (6)5: (6)3 \u003d (6)5-3 \u003d (6) 2 \u003d 6 * 6 \u003d 36. If the answer is a number in a negative degree, then such a number is converted into an ordinary fraction, in the numerator of which there is one , and in the denominator the base with the exponent obtained with the difference, only in a positive form (with a plus sign). Example 2. (2) 4: (2)6 = (2) 4-6 = (2) -2 = 1/(2)2 = ¼. The division of degrees can be written in a different form, through the fraction sign, and not as indicated in this step through the ":" sign. From this, the principle of the solution does not change, everything is done in exactly the same way, only the entry will be made with the sign of a horizontal (or oblique) fraction, instead of a colon. Example 3. (2) 4 / (2) 6 = (2) 4-6 = (2 ) -2 = 1/(2)2 = ¼.
When multiplying the same 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 the exponents differ, then most likely they can be brought to the same form by mathematical transformation. Example 6. Let it be necessary 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. Further, when raising to a power of a number. Already having a degree, the exponents are multiplied by each other: ((2)2)2: (2)3 = (2)4: (2)3 = (2) 4-3 = (2)1 = 2.
Remember, if a given base seems different from the second base, you need to look for a mathematical solution. It's just that different numbers are not given. Unless the typist made a typo in the textbook.
The power notation for a number is a shortened form of the operation of multiplying the 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 a result at the current level of technological development will not pose any difficulty.
You will need
- Internet access or Windows Calculator.
Instruction
For squaring to a power, use general rule exponentiation already having power exponent. With this operation, the indicators are multiplied, and the base remains the same. If the base is denoted as x, and the original and additional indicators as a and b, write this rule in general view you can do this: (xᵃ)ᵇ=xᵃᵇ.
Power formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.
Number c is an n-th power of a number a when:
Operations with degrees.
1. Multiplying degrees with the same base, their indicators add up:
a ma n = a m + n .
2. In the division of degrees with the same base, their indicators are subtracted:
3. The degree of the product of 2 or more factors is equal to the product of the powers of these factors:
(abc…) n = a n b n c n …
4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:
(a/b) n = a n / b n .
5. Raising a power to a power, the exponents are multiplied:
(am) n = a m n .
Each formula above is correct in the directions from left to right and vice versa.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. The root of the relationship is equal to the ratio divisible and divisor of roots:
3. When raising a root to a power, it is enough to raise the root number to this power:
4. If we increase the degree of the root in n once and at the same time raise to n th power is a root number, then the value of the root will not change:
5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:
Degree with a negative exponent. The degree of some number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to absolute value non-positive indicator:
Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.
Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
A degree with a fractional exponent. To raise a real number but to a degree m/n, you need to extract the root n th degree of m th power of this number but.
Addition and subtraction of powers
Obviously, numbers with powers can be added like other quantities , by adding them one by one 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 the same powers of the same variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is 5a 2 .
It is also obvious that if we take two squares a, or three squares a, or five squares a.
But degrees various variables And various degrees identical variables, must be added by adding them to 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 neither twice the square of a, but 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 subtrahend must be changed accordingly.
Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6
Power multiplication
Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.
So, 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 the same 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 sum degrees of terms.
So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .
Here 5 is the power of the result of the multiplication, 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 is;
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.
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 is equal to the sum or the difference of their squares.
If 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 degree.
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 divisor, or by placing them in the form of a fraction.
So a 3 b 2 divided by b 2 is 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 powers 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:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3
The rule is also valid for numbers with negative degree values.
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 the 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. Reduce exponents in $\frac $ Answer: $\frac $.
2. Reduce the exponents in $\frac$. Answer: $\frac $ or 2x.
3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to common denominator.
a 2 .a -4 is a -2 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.
degree properties
We remind you that in this lesson we understand degree properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in lessons for grade 8.
A degree with a natural indicator has several important properties, which allow you to simplify calculations in examples with powers.
Property #1
Product of powers
When multiplying powers with the same base, the base remains unchanged, and the exponents are added.
a m a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.
This property of powers also affects 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 as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17 - Present as a degree.
(0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15 - Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2 - Calculate.
Please note that in the indicated property it was 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 #2
Private degrees
When dividing powers with the same base, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.
11 3 - 2 4 2 - 1 = 11 4 = 44
Example. Solve the equation. We use the property of partial degrees.
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 degree properties.
2 11 − 5 = 2 6 = 64
Please note that property 2 dealt only with the division of 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 #3
Exponentiation
When raising a power to a power, the base of the power remains unchanged, and the exponents are multiplied.
(a n) m \u003d a n m, where "a" is any number, and "m", "n" are any natural numbers.
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.
How to multiply powers
How to multiply powers? Which powers can be multiplied and which cannot? How do you multiply a number by a power?
In algebra, you can find the product of powers in two cases:
1) if the degrees have the same basis;
2) if the degrees have the same indicators.
When multiplying powers with the same base, the base must remain the same, and the exponents must be added:
When multiplying degrees with the same indicators, the total indicator can be taken out of brackets:
Consider how to multiply powers, with specific examples.
The unit in the exponent is not written, but when multiplying the degrees, they take into account:
When multiplying, the number of degrees can be any. It should be remembered that you can not write the multiplication sign before the letter:
In expressions, exponentiation is performed first.
If you need to multiply a number by a power, you must first perform exponentiation, and only then - multiplication:
Multiplying powers with the same base
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In this lesson, we will learn how to multiply powers with the same base. First, we recall the definition of the degree and formulate a theorem on the validity of the equality . Then we give examples of its application to specific numbers and prove it. We will also apply the theorem to solve various problems.
Topic: Degree with a natural indicator and its properties
Lesson: Multiplying powers with the same bases (formula)
1. Basic definitions
Basic definitions:
n- exponent,
— n-th power of a number.
2. Statement of Theorem 1
Theorem 1. For any number but and any natural n And k equality is true:
In other words: if but- any number; n And k natural numbers, then:
Hence rule 1:
3. Explaining tasks
Output: special cases confirmed the correctness of Theorem No. 1. Let's prove it in general case, that is, for any but and any natural n And k.
4. Proof of Theorem 1
Given a number but- any; numbers n And k- natural. Prove:
The proof is based on the definition of the degree.
5. Solution of examples using Theorem 1
Example 1: Present as a degree.
To solve the following examples, we use Theorem 1.
g) 
6. Generalization of Theorem 1
Here is a generalization:
7. Solution of examples using a generalization of Theorem 1
8. Solving various problems using Theorem 1
Example 2: Calculate (you can use the table of basic degrees).
but) 
(according to the table)
b) 
Example 3: Write as a power with base 2.
but) 
Example 4: Determine the sign of the number:
, but - negative because the exponent at -13 is odd.
Example 5: Replace ( ) with a power with a base r:
We have , that is .
9. Summing up
1. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 7. 6th edition. M.: Enlightenment. 2010
1. School Assistant (Source).
1. Express as a degree:
a B C D E) 
3. Write as a power with base 2:
4. Determine the sign of the number:
but) 
5. Replace ( ) with a power of a number with a base r:
a) r 4 ( ) = r 15 ; b) ( ) r 5 = r 6
Multiplication and division of powers with the same exponents
In this lesson, we will study the multiplication of powers with the same exponents. First, let's recall the basic definitions and theorems about multiplying and dividing powers with the same bases and raising a power to a power. Then we formulate and prove theorems on multiplication and division of powers with the same exponents. And then with their help we solve the series typical tasks.
Reminder of basic definitions and theorems
Here a- base of degree
— n-th power of a number.
Theorem 1. For any number but and any natural n And k equality is true:
When multiplying powers with the same base, the exponents are added, the base remains unchanged.
Theorem 2. For any number but and any natural n And k, such that n > k equality is true:
When dividing powers with the same base, the exponents are subtracted, and the base remains unchanged.
Theorem 3. For any number but and any natural n And k equality is true:
All the above theorems were about powers with the same grounds, this lesson will consider degrees with the same indicators.
Examples for multiplying powers with the same exponents
Consider the following examples:
Let's write out the expressions for determining the degree.
Output: From the examples, you can see that 
, but this still needs to be proven. We formulate the theorem and prove it in the general case, that is, for any but And b and any natural n.
Statement and proof of Theorem 4
For any numbers but And b and any natural n equality is true:
Proof Theorem 4 .
By definition of degree:
So we have proven that .
To multiply powers with the same exponent, it is enough to multiply the bases, and leave the exponent unchanged.
Statement and proof of Theorem 5
We formulate a theorem for dividing powers with the same exponents.
For any number but And b() and any natural n equality is true:
Proof Theorem 5 .
Let's write down and by definition of degree:
Statement of theorems in words
So we have proven that .
To divide degrees with the same exponents into each other, it is enough to divide one base by another, and leave the exponent unchanged.
Solution of typical problems using Theorem 4
Example 1: Express as a product of powers.
To solve the following examples, we use Theorem 4.
To solve the following example, recall the formulas:
Generalization of Theorem 4
Generalization of Theorem 4:
Solving Examples Using Generalized Theorem 4
Continued solving typical problems
Example 2: Write as a degree of product.
Example 3: Write as a power with an exponent of 2.
Calculation Examples
Example 4: Calculate in the most rational way.
2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: VENTANA-GRAF
3. Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E. and others. Algebra 7 .M .: Education. 2006
2. School assistant (Source).
1. Present as a product of powers:
but) ; b) ; in) ; G) ;
2. Write down as the degree of the product:
3. Write in the form of a degree with an indicator of 2:
4. Calculate in the most rational way.
Mathematics lesson on the topic "Multiplication and division of powers"
Sections: Maths
Pedagogical goal:
Tasks:
Activity units of the doctrine: determination of the degree with a natural indicator; degree components; definition of private; associative law of multiplication.
I. Organization of a demonstration of mastering the existing knowledge by students. (step 1)
a) Updating knowledge:
2) Formulate a definition of the degree with a natural indicator.
a n \u003d a a a a ... a (n times)
b k \u003d b b b b a ... b (k times) Justify your answer.
II. Organization of self-assessment of the trainee by the degree of possession of relevant experience. (step 2)
Self test :( individual work in two versions.)
A1) Express the product 7 7 7 7 x x x as a power:
A2) Express as a product the degree (-3) 3 x 2
A3) Calculate: -2 3 2 + 4 5 3
I select the number of tasks in the test in accordance with the preparation of the class level.
For the test, I give a key for self-testing. Criteria: pass-fail.
III. Educational and practical task (step 3) + step 4. (the students themselves will formulate the properties)
In the course of solving problems 1) and 2), the students propose a solution, and I, as a teacher, organize a class to find a way to simplify the powers when multiplying with the same bases.
Teacher: come up with a way to simplify powers when multiplying with the same base.
An entry appears on the cluster:
The theme of the lesson is formulated. Multiplication of powers.
Teacher: come up with a rule for dividing degrees with the same bases.
Reasoning: what action checks division? a 5: a 3 = ? that a 2 a 3 = a 5
I return to the scheme - a cluster and supplement the entry - ..when dividing, subtract and add the topic of the lesson. ...and division of degrees.
IV. Communication to students of the limits of knowledge (as a minimum and as a maximum).
Teacher: the task of the minimum for today's lesson is to learn how to apply the properties of multiplication and division of powers with the same bases, and the maximum: to apply multiplication and division together.
Write on the board : a m a n = a m + n ; a m: a n = a m-n
V. Organization of the study of new material. (step 5)
a) According to the textbook: No. 403 (a, c, e) tasks with different wording
No. 404 (a, e, f) independent work, then I organize a mutual check, I give the keys.
b) For what value of m does the equality hold? a 16 a m \u003d a 32; x h x 14 = x 28; x 8 (*) = x 14
Task: come up with similar examples for division.
c) No. 417(a), No. 418 (a) Traps for students: x 3 x n \u003d x 3n; 3 4 3 2 = 9 6 ; a 16: a 8 \u003d a 2.
VI. Summarizing what has been learned, conducting diagnostic work (which encourages students, not teachers, to study this topic) (step 6)
diagnostic work.
Test(place the keys on the back of the test).
Task options: present as a degree the quotient x 15: x 3; represent as a power the product (-4) 2 (-4) 5 (-4) 7 ; for which m is the equality a 16 a m = a 32 true; find the value of the expression h 0: h 2 with h = 0.2; calculate the value of the expression (5 2 5 0) : 5 2 .
Summary of the lesson. Reflection. I divide the class into two groups.
Find the arguments of group I: in favor of knowledge of the properties of the degree, and group II - arguments that will say that you can do without properties. We listen to all the answers, draw conclusions. In subsequent lessons, you can offer statistical data and name the rubric “It doesn’t fit in my head!”
VII. Homework.
History reference. What numbers are called Fermat numbers.
P.19. #403, #408, #417
Used Books:
Properties of degrees, formulations, proofs, examples.
After the degree of the number is determined, it is logical to talk about degree properties. In this article, we will give the basic properties of the degree of a number, while touching on all possible exponents. Here we will give proofs of all properties of the degree, and also show how these properties are applied when solving examples.
Page navigation.
Properties of degrees with natural indicators
By the definition of a degree with a natural exponent, the degree of a n is the product of n factors, each of which is equal to a . Based on this definition, and using real number multiplication properties, we can obtain and justify the following properties of degree with natural exponent:
- if a>0 , then a n >0 for any natural n ;
- if a=0 , then a n =0 ;
- if a 2 m >0 , if a 2 m−1 n ;
- if m and n are natural numbers such that m>n , then for 0m n , and for a>0 the inequality a m >a n is true.
- a m a n \u003d a m + n;
- a m: a n = a m−n ;
- (a b) n = a n b n ;
- (a:b) n =a n:b n ;
- (a m) n = a m n ;
- if n is a positive integer, a and b are positive numbers, and a n n and a−n>b−n ;
- if m and n are integers, and m>n , then for 0m n , and for a>1, the inequality a m >a n is satisfied.
We immediately note that all the written equalities are identical under the specified conditions, and their right and left parts can be interchanged. For example, the main property of the fraction a m a n = a m + n with simplification of 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 definition of a degree with a natural exponent, the product of powers with the same bases of the form a m a n can be written as the product 
. Due to the properties of multiplication, the resulting expression can be written as 
, and this product is the power of a with natural exponent m+n , that is, a m+n . This completes the proof.
Let us give an example that confirms the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, according to the main property of the degree, we can write the equality 2 2 ·2 3 =2 2+3 =2 5 . Let's check its validity, for which we calculate 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 we get equal values, then the equality 2 2 2 3 =2 5 is true, and it confirms the main property of the degree.
The main 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 equality a n 1 a n 2 a n k =a n 1 +n 2 +…+n k is true.
For example, (2.1) 3 (2.1) 3 (2.1) 4 (2.1) 7 = (2.1) 3+3+4+7 =(2.1) 17 .
You can move on to the next property of degrees with a natural indicator - the property of partial 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 giving 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 it is impossible to divide by zero. The condition m>n is introduced so that we do not go beyond natural exponents. Indeed, for m>n, the exponent am−n is a natural number, otherwise it will be either zero (which happens when m−n) or a negative number (which happens when mm−n an =a (m−n) +n =am From the obtained equality am−n ·an =am and from the connection between multiplication and division it follows that am−n is a partial power of am and an This proves the property of partial powers with the same bases.
Let's take an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the considered property of the degree corresponds to the equality π 5: π 2 = π 5−3 = π 3.
Now consider product degree property: natural degree n 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 definition of a degree with a natural exponent, we have 
. Last piece based on the properties of multiplication can be rewritten as 
, which is equal to a n b n .
Here's an example: 
.
This property extends to the degree of the product of three or more factors. That is, the natural degree property 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 show this property with an example. For the product of three factors to the power of 7, we have .
The next property is natural property: the quotient of the real numbers a and b , b≠0 to the natural power n is equal to the quotient of the 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 bn =((a:b) b) n =an , and from the equality (a:b) n bn =an it follows that (a:b) n is the quotient of an to bn.
Let's write this property using the example of specific numbers: 
.
Now let's voice exponentiation property: 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 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 property in a degree is the following chain of equalities: 
.
The considered property can be extended to degree within degree within degree, and so on. For example, for any natural numbers p, q, r, and s, the equality 
. For greater clarity, let's give 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.
We start by proving the comparison property of zero and power with a natural exponent.
First, let's justify 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 allow us to assert that the result of multiplying any number of positive numbers will also be a positive number. And the power of a with natural exponent n is, by definition, 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 of a n is a positive number. By virtue of the proved property 3 5 >0 , (0.00201) 2 >0 and 
.
It is quite obvious that for any natural 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.
Let's start with the case when the exponent is an even number, denote it as 2 m , where m is a natural number. Then 
. According to the rule of multiplication of negative numbers, each of the products of the form a a is equal to the product of the modules of the numbers a and a, which means that it is a positive number. Therefore, the product will also be positive. 
and degree a 2 m . Here are examples: (−6) 4 >0 , (−2,2) 12 >0 and .
Finally, when the base of 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 negative number a results in a negative number. By virtue of this property, (−5) 3 17 n n is the product of the left and right parts of n true inequalities a properties of inequalities, the inequality being proved is of the form a n n . For example, due to this property, the inequalities 3 7 7 and 
.
It remains to prove the last of listed properties degrees with natural indicators. Let's formulate it. Of the two degrees with natural indicators and the same positive bases less than one, the degree is greater, the indicator of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree is greater, the indicator of which is greater. We turn to the proof of this property.
Let us prove that for m>n and 0m n . To do this, we write the difference a m − a n and compare it with zero. The written difference after taking a n out of brackets will take the form a n ·(a m−n −1) . The resulting product is negative as the product of a positive number an and a negative number am−n−1 (an is positive as a natural power of a positive number, and the difference am−n−1 is negative, since m−n>0 due to the initial condition m>n , whence it follows that for 0m−n it is less than one). Therefore, a m − a n m n , which was to be proved. For example, we give the correct inequality.
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 of an is a positive number, and the difference am−n−1 is a positive number, since m−n>0 by virtue of the initial condition, and for a>1 the degree of am−n is greater than one . Therefore, a m − a n >0 and a m >a n , which was to be proved. This property is illustrated by the inequality 3 7 >3 2 .
Properties of degrees with integer exponents
Since positive integers are natural numbers, then all properties of powers with positive integer exponents exactly coincide with the properties of powers with natural exponents listed and proven in the previous paragraph.
We defined a degree with a negative integer exponent, as well as a degree with a zero exponent, so that all properties of degrees with natural exponents expressed by equalities remain valid. Therefore, all these properties are valid both for zero exponents and for negative exponents, while, of course, the bases of the degrees are nonzero.
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 degrees with integer exponents:
For 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.
It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers. As an example, let's prove that the power property holds for both positive integers and nonpositive integers. To do this, we need to show that if p is zero or natural number and q is zero or a natural number, then (ap) q =ap q , (a −p) q =a (−p) q , (ap) −q =ap (−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 proved in the previous subsection. 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 .
Let us now prove that (a −p) q =a (−p) q . By definition of a degree with a negative integer exponent , then 
. By the property of the quotient in the degree, we have 
. Since 1 p =1·1·…·1=1 and , then . The last expression is, by definition, a power of the form a −(p q) , which, by virtue of the rules of multiplication, can be written as a (−p) q .
Similarly 
.
AND 
.
By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities.
In the penultimate of the properties written down, it is worth dwelling on the proof of the inequality a −n >b −n , which is true for any negative integer −n and any positive a and b for which the condition a . We write and transform the difference between the left and right parts of this inequality: 
. Since by condition a n n , therefore, b n − a n >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 a quotient of positive numbers b n − a n and a n b n . Hence, whence a −n >b −n , which was to be proved.
The last property of degrees with integer exponents is proved in the same way as the analogous property of degrees with natural exponents.
Properties of powers with rational exponents
We defined the degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, degrees with fractional exponents have the same properties as degrees with integer exponents. Namely:
- property of the product of powers with the same base
for a>0 , and if and , then for a≥0 ; - property of partial powers with the same bases
for a>0 ; - fractional product property
for a>0 and b>0 , and if and , then for a≥0 and (or) b≥0 ; - quotient property to a fractional power
for a>0 and b>0 , and if , then for a≥0 and b>0 ; - degree property in degree
for a>0 , and if and , then for a≥0 ; - the property of comparing powers with equal rational exponents: for any positive numbers a and b, a 0 the inequality a p p is valid, and for p p >b p ;
- the property of comparing degrees with rational exponents and equal grounds: for rational numbers p and q , p>q for 0p q , and for a>0 the inequality a p >a q .
- a p a q = a p + q ;
- a p:a q = a p−q ;
- (a b) p = a p b p ;
- (a:b) p =a p:b p ;
- (a p) q = a p q ;
- for any positive numbers a and b , a 0 the inequality a p p is valid, and for p p >b p ;
- for irrational numbers p and q , p>q for 0p q , and for a>0 the inequality a p >a q .
The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, on the properties of the arithmetic root of the nth degree, and on the properties of a degree with an integer exponent. Let's give proof.
By definition of the degree with a fractional exponent and , then 
. The properties of the arithmetic root allow us to write the following equalities. Further, using the degree property with an integer exponent, we obtain , whence, by the definition of a degree with a fractional exponent, we have 
, and the exponent of the degree obtained can be converted as follows: . This completes the proof.
The second property of powers with fractional exponents is proved in exactly the same way: 
The rest of the equalities are proved by similar principles: 
We turn to the proof of the next property. Let us prove that for any positive a and b , a 0 the inequality a p p is valid, and for p p >b p . We write the rational number p as m/n , where m is an integer and n is a natural number. Conditions p 0 in this case will be equivalent to conditions m 0, respectively. For m>0 and am m . From this inequality, by the property of the roots, we have , and since a and b are positive numbers, then, based on the definition of the degree with a fractional exponent, the resulting inequality can be rewritten as , that is, a p p .
Similarly, when m m >b m , whence , 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 for 0p q , and for a>0 the inequality a p >a q . We can always come to a common denominator rational numbers p and q, let us 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 the comparison rule ordinary fractions from same denominators. Then, by the property of comparing powers with the same bases and natural exponents, for 0m 1 m 2 , and for a>1, the inequality a m 1 >a m 2 . These inequalities in terms of the properties of the roots can be rewritten, respectively, as 
And 
. And the definition of a degree with a rational exponent allows us to pass to the inequalities and, respectively. From here we draw the final conclusion: for p>q and 0p q , and for a>0, the inequality a p >a q .
Properties of degrees with irrational exponents
From how the degree with irrational indicator, we can conclude that it has all the properties of powers with rational exponents. So for any a>0 , b>0 and irrational numbers p and q the following are true properties of degrees with irrational exponents:
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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Obviously, numbers with powers can be added like other quantities , by adding them one by one 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 the same powers of the same variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is 5a 2 .
It is also obvious that if we take two squares a, or three squares a, or five squares a.
But degrees various variables And various degrees identical variables, must be added by adding them to 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 neither twice the square of a, but 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 subtrahend 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
Power multiplication
Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.
So, 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 the same 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 sum degrees of terms.
So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .
Here 5 is the power of the result of the multiplication, 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 is;
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.
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 is equal to the sum or difference of their squares.
If 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 degree.
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 divisor, or by placing them in the form of a fraction.
So a 3 b 2 divided by b 2 is a 3 .
Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$
Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. 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 powers with the same base, their exponents are subtracted..
So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac(yyy)(yy) = y$.
And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.
Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3
The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.
h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$
It is necessary to master the 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. Reduce the exponents in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.
2. Reduce the exponents in $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.
3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to a common denominator.
a 2 .a -4 is a -2 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.
9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.
