Degree properties with a real exponent are examples of decision. Properties of degrees, formulations, proofs, examples. Degree with rational exponent


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.

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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:

  1. the main property of the degree a m ·a n =a m+n , its generalization ;
  2. the property of partial powers with the same bases a m:a n =a m−n ;
  3. product degree property (a b) n =a n b n , its extension ;
  4. private property in natural degree(a:b) n =a n:b n ;
  5. exponentiation (a m) n =a m n , its generalization (((a n 1) n 2) ...) n k =a n 1 n 2 ... n k;
  6. comparing degree with zero:
    • if a>0 , then a n >0 for any natural 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 integers, that m>n , then for 0 0 the inequality a m >a n is true.

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 the 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 a 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 \u003d 2 2 2 2 2 \u003d 32, since equal values ​​are obtained, then the equality 2 2 2 3 \u003d 2 5 is correct, 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.

    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 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 obtained equality a m−n ·a n =a m and from it follows that a m−n is a quotient of powers of a m and a n . 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: the natural degree n of the product of any two real numbers a and b is equal to the product of the degrees 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 . The last product, 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 power 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 b n =((a:b) b) n =a n, and the equality (a:b) n b n =a n implies that (a:b) n is the quotient of a n divided by b n .

    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, 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.

    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 . For each of the products of the form a·a is equal to the product of the modules of the numbers a and a, therefore, 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. Due to this property (−5) 3<0 , (−0,003) 17 <0 и .

    We turn to the property of comparing degrees with the same natural exponents, which has the following formulation: of two degrees with the same natural exponents, n is less than the one whose base is less, and more than the one whose base is greater. Let's prove it.

    Inequality a n properties of inequalities the inequality being proved of the form a n (2,2) 7 and .

    It remains to prove the last of the listed properties of powers with natural exponents. 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 whose indicator is greater is greater. We turn to the proof of this property.

    Let us prove that for m>n and 0 0 due to the initial condition m>n , whence it follows 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 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.

The degree with a negative integer exponent, as well as the degree with a zero exponent, we defined in such a way 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:

  1. a m a n \u003d 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 is fulfilled.

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 a natural number and q is zero or a natural number, then the equalities (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 multiplication rules, 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 . 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 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:

The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, on 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 property of the degree 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 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 и p>0 in this case will be equivalent to the conditions m<0 и m>0 respectively. For m>0 and a

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

It remains to prove the last of the listed properties. Let's 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, 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 . 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 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 this we draw the final conclusion: for p>q and 0 0 – inequality a p >a q .

Properties of degrees with irrational exponents

From how a degree with an irrational exponent is defined, it can be concluded 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 degrees 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.

Bibliography.

  • Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
  • Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
  • Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
  • Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
  • Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
  • Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

S. Shestakov,
Moscow

A written exam

Grade 11
1. Calculations. Expression conversion

§ 3. Degree with real exponent

The exercises in § 5 of the first chapter of the collection are mainly related to the exponential function and its properties. In this paragraph, as in the previous ones, not only the ability to perform transformations based on known properties is tested, but also the students' mastery of functional symbols. Among the tasks of the collection, the following groups can be distinguished:

  • exercises that test the assimilation of the definition of an exponential function (1.5.A06, 1.5.B01–B04) and the ability to use functional symbols (1.5A02, 1.5.B05, ​​1.5C11);
  • exercises on the transformation of expressions containing a power with a real exponent, and on the calculation of the values ​​of such expressions and the values ​​of the exponential function (1.5B07, 1.5B09, 1.5.C02, 1.5.C04, others);
  • exercises for comparing the values ​​of expressions containing a degree with a real exponent, requiring the use of properties of a degree with a real exponent and an exponential function (1.5.B11, 1.5C01, 1.5C12, 1.5D01, 1.5D11);
  • other exercises (including those related to positional notation of numbers, progressions, etc.) - 1.5.A03, 1.5.B08, 1.5.C06, 1.5. C09, 1.5.C10, 1.5.D07, 1.5.D09.

Consider a number of tasks related to functional symbolism.

1.5.A02. e) Functions are given

Find the value of the expression f 2 (x) - g 2 (x).

Solution. Let's use the difference of squares formula:

Answer: -12.

1.5.C11. b) Functions are given

Find the value of the expression f(x) f(y) - g(x) g(y) if f(x - y) = 9.

Here are brief solutions to exercises on transforming expressions containing a power with a real exponent, and on calculating the values ​​of such expressions and the values ​​of an exponential function.

1.5.B07. a) It is known that 6 a – 6 –a= 6. Find the value of the expression (6 a– 6) 6 a .

Solution. It follows from the conditions of the problem that 6 a – 6 = 6 -a. Then

(6 a– 6) 6a = 6 -a 6 a = 1.

1.5.C05. b) Find the value of the expression 7 a-b, if

Solution. By condition Divide the numerator and denominator of the left side of this equation by 7 b . Get

Let's make a replacement. Let y = 7 a-b. Equality takes the form

We solve the resulting equation

The next group of exercises are tasks for comparing the values ​​of expressions containing a degree with a real exponent, requiring the use of properties of a degree with a real exponent and an exponential function.

1.5.B11. b) Arrange the numbers f(60), g(45) and h(30) in descending order if f(x) = 5 x , g(x) = 7 x and h(x) = 3 x .

Solution. f(60) = 5 60 , g(45) = 7 45 and h(30) = 3 30 .

Let's transform these degrees so that we get the same indicators:

5 60 =625 15 , 7 45 =343 15 , 3 30 =9 15 .

Let's write the bases in descending order: 625 > 343 > 9.

Therefore, the required order is: f(60), g(45), h(30).

Answer: f(60), g(45), h(30).

1.5.C12. a) Compare , where x and y are some real numbers.

Solution.

That's why

That's why

Since 3 2 > 2 3 , we get that

Answer:

1.5.D11. a) Compare the numbers

Since we get

Answer:

At the end of the review of tasks for a degree with a real indicator, we will consider exercises related to the positional notation of a number, progressions, etc.

1.5.A03. b) Given a function f(x) = (0,1) x . Find the value of the expression 6f(3) + 9f(2) + 4f(1) + 4f(0).

4f(0) + 4f(1) + 9f(2) + 6f(3) = 4 1 + 4 0.1 + 9 0.01 + 6 0.001 = 4.496.

Thus, this expression is the expansion into the sum of the digit units of the decimal fraction 4.496.

Answer: 4.496.

1.5.D07. a) The function f(x) = 0.1 x is given. Find the value of the expression f 3 (1) - f 3 (2) + f 3 (3) + ... + (-1) n-1 f 3 (n) + ...

f 3 (1)–f 3 (2)+f 3 (3)+...+(–1) n–1 f 3 (n)+...= 0.1 3 –0.1 6 +0 ,1 9 +...+(–1) n–1 0.1 3n + ...

This expression is the sum of an infinitely decreasing geometric progression with the first term 0.001 and the denominator -0.001. The sum is

1.5.D09. a) Find the value of the expression 5 2x +5 2y +2 5x 5 y – 25 y 5 x if 5 x –5 y =3, x + y = 3.

5 2x +5 2y +25 x 5 y –25 y 5 x =(5 x – 5 y) 2 +2 5 x 5 y +5 x 5 y (5 x – 5 y)=3 2 +2 5 x+y +5 x+y 3=3 2 +2 5 3 +3 5 3 =634.

Answer: 634.

§ 4. Logarithmic expressions

When repeating the topic “Transformation of logarithmic expressions” (§ 1.6 of the collection), one should recall a number of basic formulas related to logarithms:

Here are a number of formulas, the knowledge of which is not required to solve problems of levels A and B, but may be useful in solving more complex problems (the number of these formulas can be either reduced or increased depending on the views of the teacher and the level of preparedness of students):

Most of the exercises from § 1.6 of the collection can be attributed to one of the following groups:

  • exercises on the direct use of the definition and properties of logarithms (1.6.A03, 1.6.A04, 1.6.B01, 1.6.B05, ​​1.6.B08, 1.6.B10, 1.6.C09, 1.6.D01, 1.6.D08, 1.6.D10);
  • exercises for calculating the value of a logarithmic expression from a given value of another expression or logarithm (1.6.C02, 1.6.C09, 1.6.D02);
  • exercises for comparing the values ​​of two expressions containing logarithms (1.6.C11);
  • exercises with a complex multi-step task (1.6.D11, 1.6.D12).

Here are brief solutions to exercises on the direct use of the definition and properties of logarithms.

1.6.B05. a) Find the value of the expression

Solution.

The expression takes the form

1.6.D08. b) Find the value of the expression (1 - log 4 36)(1 - log 9 36).

Solution. Let's use the properties of logarithms:

(1 - log 4 36)(1 - log 9 36) =

= (1 – log 4 4 – log 4 9)(1 – log 9 4 – log 9 9) =

= –log 4 9 (–log 9 4) = 1.

1.6.D10. a) Find the value of the expression

Solution. Let's transform the numerator:

log 6 42 log 7 42=(1 + log 6 7)(1 + log 7 6)=1 + log 6 7 + log 7 6 + log 6 7 log 7 6.

But log 6 7 log 7 6 = 1. So the numerator is 2 + log 6 7 + log 7 6 and the fraction is 1.

Let's move on to solving exercises for calculating the value of a logarithmic expression given the value of another expression or logarithm.

1.6.D02. a) Find the value of the expression log 70 320 if log 5 7= a, log 7 2= b.

Solution. Let's transform the expression. Let's move on to base 7:

It follows from the condition that . That's why

The following problem requires you to compare the values ​​of two expressions that contain logarithms.

1.6.C11. a) Compare the numbers

Solution. Let's take both logarithms to base 2.

Therefore, these numbers are equal.

Answer: these numbers are equal.

Independent work of a 1st year student on the topic Degrees with a valid indicator. Degree properties with real exponent (6 hours)

    Study theoretical material and make notes (2 hours)

    Solve the crossword puzzle (2 hours)

    Do homework (2 hours)

Reference and didactic material is provided below.

On the concept of degree with a rational exponent

Some of the mostcommon

Types of transcendental functions before

Totally indicative, open access to

Many studies.

L. Eiler

From the practice of solving increasingly complex algebraic problems and operating with powers, it became necessary to generalize the concept of degree and expand it by introducing zero, negative and fractional numbers as an exponent.

The equality a 0 = 1 (for ) was used in his writings at the beginning of the 15th century. Samarkand scientist al-Kashi. Regardless of him, the zero indicator was introduced by N. Shuke in the 15th century. The latter also introduced negative exponents. The idea of ​​fractional exponents is contained in the French mathematician N. Orem (XIV century) in his

work "Algorism of proportions". Instead of our sign, he wrote , instead he wrote 4. Orem verbally formulates the rules for actions with degrees, for example (in modern notation): , etc.

Later, fractional, as well as negative, exponents are found in "Complete Arithmetic" (1544) by the German mathematician M. Stiefel and S. Stevin. The latter writes that the root of the degree P from the number but can be counted as a degree but with a fraction.

The expediency of introducing zero, negative and fractional indicators and modern symbols was first written in detail in 1665 by the English mathematician John Vallis. His work was completed by I. Newton, who began to systematically apply new symbols, after which they entered into common use.

The introduction of a degree with a rational exponent is one of many examples of a generalization of the concept of a mathematical action. A degree with zero, negative, and fractional exponents is defined in such a way that the same rules of action apply to it as for a degree with a natural exponent, i.e., so that the basic properties of the originally defined concept of degree are preserved, namely:

The new definition of a degree with a rational exponent does not contradict the old definition of a degree with a natural exponent, i.e., the meaning of the new definition of a degree with a rational exponent is preserved for the particular case of a degree with a natural exponent. This principle, observed in the generalization of mathematical concepts, is called the principle of permanence (preservation, constancy). It was expressed in an imperfect form in 1830 by the English mathematician J. Peacock, and it was fully and clearly established by the German mathematician G. Hankel in 1867. The principle of permanence is also observed when generalizing the concept of a number and expanding it to the concept of a real number, and before that introduction of the concept of multiplication by a fraction, etc.

Power function andgraphicsolving equations andinequalities

Thanks to the discovery of the method of coordinates and analytical geometry, start from the 17th century. the generally applicable graphical study of functions and the graphical solution of equations became possible.

Power a function is a function of the form

where α is a constant real number. Initially, however, we restrict ourselves to rational values ​​of α and instead of equality (1) we write:

where - rational number. For and by definition, respectively, we have:

at=1, y = x.

schedule the first of these functions on the plane is a straight line parallel to the axis Oh, and the second is the bisector of the 1st and 3rd coordinate angles.

When the function graph is a parabola . Descartes, who denoted the first unknown by z, the second - through y, third - through x:, wrote the parabola equation like this: ( z- abscissa). He often used a parabola to solve equations. To solve, for example, a 4th degree equation

Descartes via substitution

got a quadratic equation with two unknowns:

depicting a circle located in one plane (zx) with parabola (4). Thus, Descartes, introducing the second unknown (X), splits equation (3) into two equations (4) and (5), each of which represents a certain locus of points. The ordinates of their intersection points give the roots of equation (3).

“One day the king decided to choose his first assistant from among his courtiers. He led everyone to a huge castle. "Whoever opens it first will be the first helper." No one even touched the castle. Only one vizier came up and pushed the lock, which opened. It was not locked.

Then the king said: “You will receive this position because you rely not only on what you see and hear, but rely on your own strength and are not afraid to make an attempt.”

And today we will try, try to come to the right decision.

1. What mathematical concept are the words associated with:

Base

Indicator (Degree)

What words can combine the words:

rational number

Integer

Natural number

Irrational number (Real number)

Formulate the topic of the lesson. (Power with real exponent)

- repeat the properties of the degree

– consider the use of degree properties in calculations and simplifications of expressions

- development of computational skills.

So, a p, where p is a real number.

Give examples (choose from expressions 5–2, , 43, ) degrees

- with a natural indicator

- with integer value

- with a rational indicator

- with an irrational indicator

For what values ​​of a does the expression make sense?

a n , where n (a is any)

a m , where m (and not equal to 0) How to go from a negative exponent to a positive exponent?

Where p, q (a > 0)

What actions (mathematical operations) can be performed with degrees?

Set match:

When multiplying powers with equal bases

The bases are multiplied, but the exponent remains the same

When dividing powers with equal bases

The bases are divided, but the exponent remains the same


In this article, we will understand what is degree of. Here we will give definitions of the degree of a number, while considering in detail all possible exponents of the degree, starting with a natural exponent, ending with an irrational one. In the material you will find a lot of examples of degrees covering all the subtleties that arise.

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Degree with natural exponent, square of a number, cube of a number

Let's start with . Looking ahead, let's say that the definition of the degree of a with natural exponent n is given for a , which we will call base of degree, and n , which we will call exponent. Also note that the degree with a natural indicator is determined through the product, so to understand the material below, you need to have an idea about the multiplication of numbers.

Definition.

Power of number a with natural exponent n is an expression of the form a n , whose value is equal to the product of n factors, each of which is equal to a , that is, .
In particular, the degree of a number a with exponent 1 is the number a itself, that is, a 1 =a.

Immediately it is worth mentioning the rules for reading degrees. The universal way to read the entry a n is: "a to the power of n". In some cases, such options are also acceptable: "a to the nth power" and "nth power of the number a". For example, let's take the power of 8 12, this is "eight to the power of twelve", or "eight to the twelfth power", or "twelfth power of eight".

The second power of a number, as well as the third power of a number, have their own names. The second power of a number is called the square of a number, for example, 7 2 is read as "seven squared" or "square of the number seven". The third power of a number is called cube number, for example, 5 3 can be read as "five cubed" or say "cube of the number 5".

It's time to bring examples of degrees with physical indicators. Let's start with the power of 5 7 , where 5 is the base of the power and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9 .

Please note that in the last example, the base of the degree 4.32 is written in brackets: to avoid discrepancies, we will take in brackets all the bases of the degree that are different from natural numbers. As an example, we give the following degrees with natural indicators , their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity at this point, we will show the difference contained in the records of the form (−2) 3 and −2 3 . The expression (−2) 3 is the power of −2 with natural exponent 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .

Note that there is a notation for the degree of a with an exponent n of the form a^n . Moreover, if n is a multivalued natural number, then the exponent is taken in brackets. For example, 4^9 is another notation for the power of 4 9 . And here are more examples of writing degrees using the “^” symbol: 14^(21) , (−2,1)^(155) . In what follows, we will mainly use the notation of the degree of the form a n .

One of the problems, the reverse of exponentiation with a natural exponent, is the problem of finding the base of the degree from a known value of the degree and a known exponent. This task leads to .

It is known that the set of rational numbers consists of integers and fractional numbers, and each fractional number can be represented as a positive or negative ordinary fraction. We defined the degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of the degree with a rational exponent, we need to give the meaning of the degree of the number a with a fractional exponent m / n, where m is an integer and n is a natural number. Let's do it.

Consider a degree with a fractional exponent of the form . In order for the property of degree in a degree to remain valid, the equality must hold . If we take into account the resulting equality and the way we defined , then it is logical to accept, provided that for given m, n and a, the expression makes sense.

It is easy to check that all properties of a degree with an integer exponent are valid for as (this is done in the section on the properties of a degree with a rational exponent).

The above reasoning allows us to make the following output: if for given m, n and a the expression makes sense, then the power of the number a with a fractional exponent m / n is the root of the nth degree of a to the power m.

This statement brings us close to the definition of a degree with a fractional exponent. It remains only to describe for which m, n and a the expression makes sense. Depending on the restrictions imposed on m , n and a, there are two main approaches.

    The easiest way to constrain a is to assume a≥0 for positive m and a>0 for negative m (because m≤0 has no power of 0 m). Then we get the following definition of the degree with a fractional exponent.

    Definition.

    Power of a positive number a with fractional exponent m/n, where m is an integer, and n is a natural number, is called the root of the nth of the number a to the power of m, that is, .

    The fractional degree of zero is also defined with the only caveat that the exponent must be positive.

    Definition.

    Power of zero with fractional positive exponent m/n, where m is a positive integer and n is a natural number, is defined as .
    When the degree is not defined, that is, the degree of the number zero with a fractional negative exponent does not make sense.

    It should be noted that with such a definition of the degree with a fractional exponent, there is one nuance: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0 . For example, it makes sense to write or , and the above definition forces us to say that degrees with a fractional exponent of the form are meaningless, since the base must not be negative.

    Another approach to determining the degree with a fractional exponent m / n is to separately consider the even and odd exponents of the root. This approach requires an additional condition: the degree of the number a, whose exponent is , is considered the degree of the number a, the exponent of which is the corresponding irreducible fraction (the importance of this condition will be explained below). That is, if m/n is an irreducible fraction, then for any natural number k the degree is first replaced by .

    For even n and positive m, the expression makes sense for any non-negative a (root even degree from negative number does not make sense), for negative m the number a must still be different from zero (otherwise it will be division by zero). And for odd n and positive m, the number a can be anything (the root of an odd degree is defined for any real number), and for negative m, the number a must be different from zero (so that there is no division by zero).

    The above reasoning leads us to such a definition of the degree with a fractional exponent.

    Definition.

    Let m/n be an irreducible fraction, m an integer, and n a natural number. For any contraction common fraction degree is replaced by . The power of a with an irreducible fractional exponent m / n is for

    Let us explain why a degree with a reducible fractional exponent is first replaced by a degree with an irreducible exponent. If we simply defined the degree as , and did not make a reservation about the irreducibility of the fraction m / n , then we would encounter situations similar to the following: since 6/10=3/5 , then the equality , but , but .

Lesson topic: Degree with rational and real exponents.

Goals:

    Educational :

    • generalize the concept of degree;

      to develop the ability to find the value of the degree with a real indicator;

      consolidate the ability to use the properties of the degree when simplifying expressions;

      develop the skill of using the properties of the degree in calculations.

    Educational :

    • intellectual, emotional, personal development student;

      develop the ability to generalize, systematize on the basis of comparison, draw a conclusion;

      activate independent activity;

      develop curiosity.

    Educational :

    • education of communicative and informational culture of students;

      aesthetic education is carried out through the formation of the ability to rationally, accurately draw up a task on a blackboard and in a notebook.

Students should know: definition and properties of degree with real exponent

Students should be able to:

    determine whether an expression with a degree makes sense;

    use the properties of the degree in calculations and simplification of expressions;

    solve examples containing a degree;

    compare, find similarities and differences.

Lesson form: seminar - workshop, with elements of research. Computer support.

Form of organization of training: individual, group.

Pedagogical technologies : problem learning, learning in collaboration, personally - oriented learning, communicative.

Lesson type: lesson of research and practical work.

Lesson visuals and handouts:

    presentation

    formulas and tables (application 1.2)

    assignment for independent work (Appendix 3)

Lesson Plan

№

Lesson stage

Purpose of the stage

Time, min.

Lesson start

Reporting the topic of the lesson, setting the goals of the lesson.

1-2 min

oral work

Review the power formulas.

Degree properties.

4-5 min.

Frontal solution

boards from textbook No. 57 (1,3,5)

№58(1,3,5) with detailed adherence to the solution plan.

Formation of skills and abilities

have students apply properties

degrees when finding the values ​​of the expression.

8-10 min.

Work in microgroups.

Identifying gaps in knowledge

students, creating conditions for

individual development student

on the lesson.

15-20 min.

Summing up the work.

Track the success of your work

Students, when independently solving problems on a topic, find out

the nature of the difficulties, their causes,

provide collective solutions.

5-6 min.

Homework

Introduce students to homework. Give the necessary explanations.

1-2 min.

DURING THE CLASSES

Organizing time

Hello guys! Write in your notebooks the number, the topic of the lesson.

They say that the inventor of chess, as a reward for his invention, asked the raja for some rice: he asked to put one grain on the first cell of the board, on the second - 2 times more, i.e. 2 grains, on the third - 2 more times more, i.e. 4 grains, etc. up to 64 cells.

His request seemed too modest to the Raja, but it soon became clear that it was impossible to fulfill it. The number of grains that had to be given to the inventor of chess as a reward is expressed by the sum

1+2+2 2 +2 3 +…+2 63 .

This amount is equal to a huge number

18446744073709551615

And it is so large that this amount of grain could cover the entire surface of our planet, including the world ocean, with a layer of 1 cm.

Degrees are used when writing numbers and expressions, which makes them more compact and convenient for performing actions.

Degrees are often used to measure physical quantities, which can be "very large" or "very small".

The mass of the Earth 6000000000000000000000t is written as a product of 6.10 21 T

The diameter of a water molecule 0.0000000003 m is written as a product

3.10 -10 m.

1. With which mathematical concept related words:

Base
Indicator(Degree)


What words can combine the words:
rational number
Integer
Natural number
irrational number(Real number)
Formulate the topic of the lesson.(Power with real exponent)

2. So a x,wherex is a real number. Select from expressions

With natural indicator

With an integer

FROM rational indicator

FROM irrational indicator

3. What is our goal?(USE)
What kindobjectives of our lesson ?
- Generalize the concept of degree.

Tasks:

– repeat the properties of the degree
– consider the use of degree properties in calculations and simplifications of expressions
– development of computational skills

4 . Degree with rational exponent

Base

degree

Degree with exponentr, base a (nN, mn

r= n

r= - n

r= 0

r= 0

r=0

a n= a. a. … . a

a -n=

a 0 =1

a n=a.a. ….a

a -n=

Does not exist

Does not exist

a 0 =1

a=0

0 n=0

Does not exist

Does not exist

Does not exist

5 . From these expressions, choose those that do not make sense:

6 . Definition

If numberr- natural, then rthere is a workrnumbers, each of which is equal to a:

a r= a. a. … . a

If numberr- fractional and positive, that is, wheremAndn- natural

numbers, then

If the indicatorris rational and negative, then the expressiona r

is defined as the reciprocal ofa - r

or

If

7 . For example

8 . Powers of positive numbers have the following basic properties:

9 . Calculate

10. What actions (mathematical operations) can be performed with degrees?

Set match:

A) When multiplying powers with equal grounds

1) The bases are multiplied, but the exponent remains the same

B) When dividing degrees with equal bases

2) The bases are divided, but the exponent remains the same

B) When raising a power to a power

3) The base remains the same, but the exponents are multiplied

D) When multiplying powers with equal exponents

4) The base remains the same, and the exponents are subtracted

E) When dividing degrees with equal indicators

5) The base remains the same, and the indicators add up

11 . From the textbook (at the blackboard)

For a class solution:

№57 (1,3,5)

№58 (1, 3, 5)

№59 (1, 3)

№60 (1,3)

12 . By USE materials

(independent work) on leaflets

XIVcentury.

Answer: Oresma. 13. Additionally (individually) for those who can complete the tasks faster:

14. Homework

§ 5 (know definitions, formulas)

№57 (2, 4, 6)

№58 (2,4)

№59 (2,4)

№60 (2,4) .

At the end of the lesson:

“Mathematics already then needs to be taught, that it puts the mind in order”

– So said the great Russian mathematician Mikhail Lomonosov.


- Thank you for the lesson!

Attachment 1

1. Degrees. Basic properties

indicator

a 1 =a

a n=a.a. ….a

aRn

3 5 =3 . 3 . 3 . 3 . 3 . 3=243,

(-2) 3 =(-2) . (-2) . (-2)= - 8

Degree with integer exponent

a 0 =1,

where a

0 0 - not defined.

Degree with rational

indicator

wherea

m n

Degree with irrational exponent

Answer: ==25.9...

1. a x. a y=a x+y

2.a x: a y==a x-y

3. .(a x) y=a x.y

4.(a.b) n=a n.b n

5. (=

6. (

Appendix 2

2. Degree with a rational exponent

Base

degree

Degree with exponentr, base a (nN, mn

r= n

r= - n

r= 0

r= 0

r=0

a n= a. a. … . a

a -n=

a 0 =1

a n=a.a. ….a

a -n=

Does not exist

Does not exist

a 0 =1

a=0

0 n=0

Does not exist

Does not exist

Does not exist

Appendix 3

3. Independent work

For the first time, actions on powers were used by a French mathematicianXIVcentury.

Decipher the name of the French scientist.

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