Examples of operations with powers with rational exponents. Number power: definitions, notation, examples

MBOU "Sidorskaya"

comprehensive school"

Development of an open lesson plan

in algebra in 11th grade on the topic:

Prepared and carried out

math teacher

Iskhakova E.F.

Outline of an open lesson in algebra in 11th grade.

Subject : “A degree with a rational exponent.”

Lesson type : Learning new material

Lesson Objectives:

Introduce students to the concept of a degree with a rational exponent and its basic properties, based on previously studied material (degree with an integer exponent).

Develop computational skills and the ability to convert and compare numbers with rational exponents.

To develop mathematical literacy and mathematical interest in students.

Equipment : Task cards, student presentation by degree with an integer indicator, teacher presentation by degree with a rational indicator, laptop, multimedia projector, screen.

During the classes:

Organizing time.

Checking the mastery of the covered topic using individual task cards.

Task No. 1.

=2;

B) =x + 5;

Solve the system of irrational equations: - 3 = -10,

4 - 5 =6.

Task No. 2.

Solve the irrational equation: = - 3;

B) = x - 2;

Solve the system of irrational equations: 2 + = 8,

3 - 2 = - 2.

Communicate the topic and objectives of the lesson.

The topic of our lesson today is “ Power with rational exponent».

Explanation of new material using the example of previously studied material.

You are already familiar with the concept of a degree with an integer exponent. Who will help me remember them?

Repetition using presentation " Degree with an integer exponent».

For any numbers a, b and any integers m and n the equalities are valid:

a m * a n =a m+n ;

a m: a n =a m-n (a ≠ 0);

(a m) n = a mn ;

(a b) n =a n * b n ;

(a/b) n = a n /b n (b ≠ 0) ;

a 1 =a ; a 0 = 1(a ≠ 0)

Today we will generalize the concept of power of a number and give meaning to expressions that have a fractional exponent. Let's introduce definition degrees with a rational exponent (Presentation “Degree with a rational exponent”):

Power of a > 0 with rational exponent r = , Where m is an integer, and n – natural ( n > 1), called the number m .

So, by definition we get that = m .

Let's try to apply this definition when completing a task.

EXAMPLE No. 1

I Present the expression as a root of a number:

A) B) IN) .

Now let's try to apply this definition in reverse

II Express the expression as a power with a rational exponent:

A) 2 B) IN) 5 .

The power of 0 is defined only for positive exponents.

0 r= 0 for any r> 0.

Using this definition, Houses you will complete #428 and #429.

Let us now show that with the definition of a degree with a rational exponent formulated above, the basic properties of degrees are preserved, which are true for any exponents.

For any rational numbers r and s and any positive a and b, the following equalities hold:

1 0 . a r a s =a r+s ;

EXAMPLE: *

20 . a r: a s =a r-s ;

EXAMPLE: :

3 0 . (a r ) s =a rs ;

EXAMPLE: ( -2/3

4 0 . ( ab) r = a r b r ; 5 0 . ( = .

EXAMPLE: (25 4) 1/2 ; ( ) 1/2

EXAMPLE of using several properties at once: * : .

Physical education minute.

We put the pens on the desk, straightened the backs, and now we reach forward, we want to touch the board. Now we’ve raised it and leaned right, left, forward, back. You showed me your hands, now show me how your fingers can dance.

Working on the material

Let us note two more properties of powers with rational exponents:

6 0 . Let r is a rational number and 0< a < b . Тогда

a r < b r at r> 0,

a r < b r at r< 0.

7 0 . For any rational numbersr And s from inequality r> s follows that

a r>a r for a > 1,

a r < а r at 0< а < 1.

EXAMPLE: Compare the numbers:

AND ; 2 300 and 3 200 .

Lesson summary:

Today in the lesson we recalled the properties of a degree with an integer exponent, learned the definition and basic properties of a degree with a rational exponent, and examined the application of this theoretical material in practice when performing exercises. I would like to draw your attention to the fact that the topic “Degree with a rational exponent” is mandatory in the Unified State Examination tasks. When preparing homework ( No. 428 and No. 429

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:

the main property of the degree a m ·a n =a m+n, its generalization;
property of quotient powers with identical bases a m:a n =a m−n ;
product power property (a·b) n =a n ·b n , its extension;
property of the quotient to the natural degree (a:b) n =a n:b n ;
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;
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 ;
if a and b are positive numbers and a
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 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:

a m ·a n =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 b−n ;

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:

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 b p ;

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 textbook for 5th grade. educational institutions.

Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.

Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.

Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.

Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.

Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

From integer exponents of the number a, the transition to rational exponents suggests itself. Below we will define a degree with a rational exponent, and we will do this in such a way that all the properties of a degree with an integer exponent are preserved. This is necessary because integers are part of the rational numbers.

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

Let's consider a degree with a fractional exponent of the form . For the power-to-power property to remain valid, the equality must hold . If we take into account the resulting equality and how we determined the nth root of the degree, then it is logical to accept, provided that given the given m, n And a the expression makes sense.

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

The above reasoning allows us to make the following conclusion: if given data m, n And a the expression makes sense, then the power of the number a with a fractional indicator m/n called the root n th degree of a to a degree m.

This statement brings us close to the definition of a degree with a fractional exponent. All that remains is to describe at what m, n And a the expression makes sense. Depending on the restrictions imposed on m, n And a There are two main approaches.

1. The easiest way is to impose a restriction on a, having accepted a≥0 for positive m And a>0 for negative m(since when m≤0 degree 0 m not determined). Then we get the following definition of a degree with a fractional exponent.

Definition.

Power of a positive number a with a fractional indicator m/n , Where m- whole, and n– a natural number, called a root n-th of the number a to a degree m, that is, .

The fractional power of zero is also determined with the only caveat that the indicator must be positive.

Definition.

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

It should be noted that with this definition of a degree with a fractional exponent, there is one caveat: for some negative a and some m And n the expression makes sense, but we discarded these cases by introducing the condition a≥0. For example, the entries make sense or , and the definition given above forces us to say that powers with a fractional exponent of the form do not make sense, since the base should not be negative.

2. Another approach to determining the degree with a fractional exponent m/n consists in separately considering even and odd exponents of the root. This approach requires an additional condition: the power of the number a, the exponent of which is a reducible ordinary fraction, is considered a power of the number a, the indicator 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 degree is preliminarily replaced by .

For even n and positive m the expression makes sense for any non-negative a(an even root of a negative number has no meaning), for negative m number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m number a can be any (an odd root is defined for any real number), and for negative m number a must be non-zero (so that there is no division by zero).

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

Definition.

Let m/n– irreducible fraction, m- whole, and n- natural number. For any reducible fraction, the degree is replaced by . Degree of a with an irreducible fractional exponent m/n- it is for

o any real number a, whole positive m and odd natural n, For example, ;

o any non-zero real number a, negative integer m and odd n, For example, ;

o any non-negative number a, whole positive m and even n, For example, ;

o any positive a, negative integer m and even n, For example, ;

o in other cases, the degree with a fractional indicator is not determined, as for example the degrees are not defined .a we do not attach any meaning to the entry; we define the power of the number zero for positive fractional exponents m/n How , for negative fractional exponents the power of the number zero is not determined.

In conclusion of this point, let us draw attention to the fact that a fractional exponent can be written as a decimal fraction or a mixed number, for example, . To calculate the values of expressions of this type, you need to write the exponent in the form of an ordinary fraction, and then use the definition of the exponent with a fractional exponent. For the above examples we have And

The expression a n (power with an integer exponent) will be defined in all cases, except for the case when a = 0 and n is less than or equal to zero.

Properties of degrees

Basic properties of degrees with an integer exponent:

a m *a n = a (m+n) ;

a m: a n = a (m-n) (with a not equal to zero);

(a m) n = a (m*n) ;

(a*b) n = a n *b n ;

(a/b) n = (a n)/(b n) (with b not equal to zero);

a 0 = 1 (with a not equal to zero);

These properties will be valid for any numbers a, b and any integers m and n. It is also worth noting the following property:

If m>n, then a m > a n, for a>1 and a m

We can generalize the concept of the degree of a number to cases where rational numbers act as the exponent. At the same time, I would like all of the above properties to be fulfilled, or at least some of them.

For example, if the property (a m) n = a (m*n) were satisfied, the following equality would hold:

(a (m/n)) n = a m .

This equality means that the number a (m/n) must be the nth root of the number a m.

The power of some number a (greater than zero) with a rational exponent r = (m/n), where m is some integer, n is some natural number greater than one, is the number n√(a m). Based on the definition: a (m/n) = n√(a m).

For all positive r, the power of zero will be determined. By definition, 0 r = 0. Note also that for any integer, any natural m and n, and positive A the following equality is true: a (m/n) = a ((mk)/(nk)) .

For example: 134 (3/4) = 134 (6/8) = 134 (9/12).

From the definition of a degree with a rational exponent it directly follows that for any positive a and any rational r the number a r will be positive.

Basic properties of a degree with a rational exponent

For any rational numbers p, q and any a>0 and b>0 the following equalities are true:

1. (a p)*(a q) = a (p+q) ;

2. (a p):(b q) = a (p-q) ;

3. (a p) q = a (p*q) ;

4. (a*b) p = (a p)*(b p);

5. (a/b) p = (a p)/(b p).

These properties follow from the properties of the roots. All these properties are proven in a similar way, so we will limit ourselves to proving only one of them, for example, the first (a p)*(a q) = a (p + q) .

Let p = m/n, and q = k/l, where n, l are some natural numbers, and m, k are some integers. Then you need to prove that:

(a (m/n))*(a (k/l)) = a ((m/n) + (k/l)) .

First, let's bring the fractions m/n k/l to a common denominator. We get the fractions (m*l)/(n*l) and (k*n)/(n*l). Let's rewrite the left side of the equality using these notations and get:

(a (m/n))*(a (k/l)) = (a ((m*l)/(n*l)))*(a ((k*n)/(n*l)) ).

(a (m/n))*(a (k/l)) = (a ((m*l)/(n*l)))*(a ((k*n)/(n*l)) ) = (n*l)√(a (m*l))*(n*l)√(a (k*n)) = (n*l)√((a (m*l))*(a (k*n))) = (n*l)√(a (m*l+k*n)) = a ((m*l+k*n)/(n*l)) = a ((m /n)+(k/l)) .