What is a rational exponent degree. Lesson “Degree with a rational exponent. Converting expressions with roots and powers

MBOU "Sidorskaya

comprehensive school"

Development of an open lesson outline

in algebra in grade 11 on the topic:

Prepared and conducted

math teacher

Iskhakova E.F.

Outline of an open lesson in algebra in grade 11.

Topic : "Degree with a rational exponent."

Lesson type : Learning new material

Lesson objectives:

To acquaint students with the concept of a degree with a rational indicator and its main properties, based on previously studied material (a degree with a whole indicator).

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

To foster math literacy and math interest in students.

Equipment : Task cards, student's presentation by degree with a whole indicator, teacher's presentation by degree with a rational indicator, laptop, multimedia projector, screen.

During the classes:

Organizing time.

Checking the assimilation of the passed topic according to individual task cards.

Task number 1.

=2;

B) = x + 5;

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

4 - 5 =6.

Task number 2.

Solve the irrational equation: = - 3;

B) = x - 2;

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

3 - 2 = - 2.

Communication of the topic and objectives of the lesson.

The topic of our today's lesson “ Rational grade».

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

You are already familiar with the concept of a degree with a whole exponent. Who can help me remember them?

Repetition with a presentation " Integer degree».

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

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 the power of a number and give meaning to expressions that have a fractional exponent. Let's introduce definition degrees with a rational indicator (Presentation "Degree with a rational indicator"):

The power of the number a > 0 with rational exponent r = , where m Is an integer, and n - natural ( n > 1) is called the number m .

So, by definition, we get that = m .

Let's try to apply this definition when performing an assignment.

EXAMPLE # 1

I Imagine as a root of a number the expression:

A) B) V) .

Now let's try to apply this definition in reverse.

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

A) 2 B) V) 5 .

The power of the number 0 is defined only for positive indicators.

0 r= 0 for any r> 0.

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

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

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: *

twenty . 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 for the use of several properties at once: * : .

Physical education.

We put the pens on the desk, straightened the backs, and now we stretch forward, we want to touch the board. And now we raised and lean to the right, left, forward, backward. They showed me the pens, and now show me how your fingers can dance.

Work on the material

We note two more properties of degrees 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 remembered the properties of a degree with a whole exponent, learned the definition and basic properties of a degree with a rational exponent, 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 indicator" is mandatory in the USE assignments. When preparing homework ( No. 428 and No. 429

Rational grade

Khasyanova T.G.,

math teacher

The presented material will be useful for teachers of mathematics when studying the topic "Degree with a rational indicator".

The purpose of the presented material: disclosure of my experience in conducting a lesson on the topic "Degree with a rational indicator" of the work program of the discipline "Mathematics".

The methodology of the lesson corresponds to its type - a lesson in the study and primary consolidation of new knowledge. Basic knowledge and skills were updated on the basis of previous experience; primary memorization, consolidation and application of new information. The consolidation and application of the new material took place in the form of solving problems of varying complexity tested by me, giving a positive result of mastering the topic.

At the beginning of the lesson, I set the following goals for the students: educational, developmental, educational. In the lesson, I used various methods of activity: frontal, individual, pair, independent, test. The tasks were differentiated and made it possible to identify, at each stage of the lesson, the degree of assimilation of knowledge. The volume and complexity of the tasks corresponds to the age characteristics of the students. From my experience, homework, similar to the tasks solved in the classroom, allows you to reliably consolidate the acquired knowledge and skills. At the end of the lesson, reflection was carried out and the work of individual students was assessed.

The goals were achieved. The students studied the concept and properties of a degree with a rational indicator, learned to use these properties in solving practical problems. For independent work, grades are announced in the next lesson.

I believe that the methodology used by me for conducting classes in mathematics can be applied by teachers of mathematics.

Lesson Topic: Degree with Rational Score

The purpose of the lesson:

Revealing the level of mastering by students of a complex of knowledge and skills and, on its basis, the use of certain solutions to improve the educational process.

Lesson Objectives:

Educational: to form new knowledge among students of basic concepts, rules, laws for determining the degree with a rational indicator, the ability to independently apply knowledge in standard conditions, in changed and non-standard conditions;

developing: think logically and realize creative abilities;

educating: to form an interest in mathematics, to replenish the vocabulary with new terms, to obtain additional information about the world around. To cultivate patience, perseverance, the ability to overcome difficulties.

Organizing time

Updating basic knowledge

When multiplying degrees with the same bases, the indicators are added, and the base remains the same:

For instance,

2. When dividing degrees with the same bases, the exponents are subtracted, and the base remains the same:

For instance,

3. When raising a degree to a power, the exponents are multiplied, and the base remains the same:

For instance,

4. The degree of the product is equal to the product of the degrees of the factors:

For instance,

5. The degree of the quotient is equal to the quotient of the powers of the divisor n of the divisor:

For instance,

Exercises with solutions

Find the value of an expression:

Solution:

In this case, none of the properties of a degree with a natural exponent can be applied in an explicit form, since all degrees have different grounds. Let's write some degrees in a different form:

(the degree of the product is equal to the product of the degrees of the factors);

(when multiplying degrees with the same bases, the indicators are added, but the base remains the same, when raising a degree to a power, the exponents are multiplied, and the base remains the same).

Then we get:

In this example, the first four properties of the degree with a natural exponent were used.

Arithmetic square root
is a non-negative number whose square isa,
... At
- expression
not defined, because there is no real number whose square is equal to a negative numbera.

Mathematical dictation(8-10 min.)

Option

II. Option

1.Find the value of an expression

a)

b)

1.Find the value of an expression

a)

b)

2.Calculate

a)

b)

V)

2.Calculate

a)

b)

v)

Self-test(on the lapel board):

Answer matrix:

№ options / assignments

Problem 1

Task 2

Option 1

a) 2

b) 2

a) 0.5

b)

v)

Option 2

a) 1.5

b)

a)

b)

at 4

II. Formation of new knowledge

Consider the meaning of the expression, where - positive number- fractional number and m-integer, n-natural (n ›1)

Definition: the degree of a number a ›0 with a rational exponentr = , m-whole, n-natural ( n›1) is the number.

So:

For instance:

Notes:

1. For any positive a and any rational r, the number positively.

2. When
rational degree of numberanot defined.

Expressions such as
don't make sense.

3.If fractional positive number then
.

If fractional negative number, then -doesn't make sense.

For instance: - doesn't make sense.

Consider the properties of a degree with a rational exponent.

Let a> 0, в> 0; r, s are any rational numbers. Then a degree with any rational exponent has the following properties:

1.
2.
3.
4.
5.

III. Anchoring. Formation of new skills and abilities.

Task cards work in small groups in the form of a test.

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

Page navigation.

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

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

Definition.

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

It should be said right away about the rules for reading degrees. The universal way to read a record a n is as follows: "a to the power of n". In some cases, the following options are also acceptable: "a to the n-th power" and "n-th power of the number a". For example, let's take the power of 8 12, which is “eight to the power of twelve”, or “eight to the twelfth power”, or “the twelfth power of eight”.

The second degree of a number, as well as the third degree of a number, have their own names. The second power of a number is called by the square of the number for example, 7 2 reads “seven squared” or “the square of the number seven”. The third power of a number is called cube numbers for example, 5 3 can be read as “cube of five” or “cube of number 5”.

It's time to lead examples of degrees with natural values... Let's start with the exponent 5 7, here 5 is the base of the exponent 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.

Note that in the last example, the base of the 4.32 degree is written in parentheses: to avoid confusion, we will put in parentheses all 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 in this moment, we will show the difference between the entries of the form (−2) 3 and −2 3. The expression (−2) 3 is the power of −2 with a natural exponent of 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 number a with exponent n of the form a ^ n. Moreover, if n is a multivalued natural number, then the exponent is taken in parentheses. For example, 4 ^ 9 is another notation for the power of 4 9. And here are some more examples of writing degrees using the "^" symbol: 14 ^ (21), (−2,1) ^ (155). In what follows, we will mainly use the notation for the degree of the form a n.

One of the tasks, the inverse of raising to a power with a natural exponent, is the problem of finding the base of a 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 a degree with a rational exponent, we need to give a sense to the degree of a 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. For the property of degree to degree to be valid, the equality ... If we take into account the obtained equality and the way we determined it, then it is logical to accept, provided that for the given m, n and a, the expression makes sense.

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

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

This statement brings us very close to determining the degree with a fractional exponent. It remains only to describe for which m, n and a the expression makes sense. There are two main approaches depending on the constraints on m, n and a.

The easiest way is to restrict a by assuming a≥0 for positive m and a> 0 for negative m (since for m≤0 the degree 0 m is not defined). Then we get the following definition of a fractional exponent.

Definition.

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

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

Definition.

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

It should be noted that with such a definition of a 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 definition given above forces us to say that degrees with a fractional exponent of the form do not make sense, since the base should not be negative.

Another approach to determining the exponent with a fractional exponent m / n is to consider separately the odd and even exponents of the root. This approach requires an additional condition: the degree of the number a, the indicator of which is, is considered the 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, the 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 does not make sense), for negative m, the number a must still be nonzero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be nonzero (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 cancellable fraction, the exponent is replaced by. The power of a number with an irreducible fractional exponent m / n is for



Let us explain why a degree with a reducible fractional exponent is previously 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 be faced with situations similar to the following: since 6/10 = 3/5, then the equality should hold , but , a .

The expression a n (a degree 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.

Power properties

The main properties of degrees with a whole exponent:

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

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

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

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

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

It is possible to generalize the concept of the degree of a number to cases when rational numbers act as an exponent. At the same time, we 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) was 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 degree of the number zero will be determined. By definition, 0 r = 0. Note also that for any integer, any natural numbers 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).

The definition of a degree with a rational exponent directly implies the fact 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 hold:

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 proved in a similar way, so we restrict ourselves to proving only one of them, for example, the first (a p) * (a q) = a (p + q).

Let p = m / n, a 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, we bring the fractions m / n k / l to a common denominator. We get the fractions (m * l) / (n * l) and (k * n) / (n * l). We rewrite the left-hand side of the equality using these notation 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)).

After the degree of the number has been determined, it is logical to talk about properties degree... 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 natural exponents

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

1. the main property of the degree a m · a n = a m + n, its generalization;
2. property of private degrees 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. property of the quotient in natural degree (a: b) n = a n: b n;
5. raising a power to a power (a m) n = a mn, its generalization (((a n 1) n 2)…) n k = a n 1 n 2… n k;
6. comparing degree to 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 natural numbers such that m> n, then for 0 0 the inequality a m> a n is true.

Note right away that all the equalities written down are identical subject to the specified conditions, and their right and left parts can be swapped. For example, the main property of the fraction a m a n = a m + n for simplification of expressions often used as a m + n = a m a n.

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

Let's start with the property of a product of two degrees 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 degrees 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 the number a with natural exponent m + n, that is, a m + n. This completes the proof.

Let's give an example that confirms the main property of the degree. Take degrees with the same bases 2 and natural degrees 2 and 3, according to the basic property of the degree, we can write the equality 2 2 · 2 3 = 2 2 + 3 = 2 5. Let us check its validity, for which we calculate the values ​​of the expressions 2 2 · 2 3 and 2 5. 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 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 degrees with the same bases and natural exponents. So for any number k 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 instance, (2.1) 3 (2.1) 3 (2.1) 4 (2.1) 7 = (2,1) 3+3+4+7 =(2,1) 17 .

You can go to the next property of degrees with a natural exponent - property of private degrees with the same bases: for any nonzero real number a and arbitrary natural numbers m and n satisfying the condition m> n, the equality a m is true: a n = a m − n.

Before proving this property, let us discuss the meaning of 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 one 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 when 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 a m and a n. This proves the property of private degrees with the same bases.

Let's give an example. 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 powers of 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 ... Based on the properties of multiplication, the last product can be rewritten as , which is equal to a n · b n.

Let's give an example: .

This property applies to the degree of the product of three or more factors. That is, the property of the 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 by an example. For the product of three factors to the power of 7, we have.

The next property is private property in kind: the quotient of real numbers a and b, b ≠ 0 in natural power n is equal to the quotient of powers of 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 dividing a n by b n.

Let's write this property using the example of specific numbers: .

Now we will sound exponentiation property: for any real number a and any natural numbers m and n, the degree of a m to the power 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 property of degree to degree is the following chain of equalities: .

The considered property can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r, and s, the equality ... For clarity, here's 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 with proving the property of comparing zero and degree with natural exponent.

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

The product of two positive numbers is a positive number, which follows from the definition of multiplication. This fact and the properties of multiplication make it possible to assert that the result of multiplying any number of positive numbers will also be a positive number. And the degree of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These considerations allow us to assert that for any positive base a, the degree 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 for 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.

Moving on to negative bases of the degree.

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 absolute values ​​of the numbers a and a, which means that it is a positive number. Therefore, the product and the degree a 2 m. Here are some examples: (−6) 4> 0, (−2,2) 12> 0 and.

Finally, when the base of the exponent a is negative 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 multiplying it 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 indicators, which has the following formulation: of two degrees with the same natural indicators, n is less than the one whose base is less, and the greater is the one whose base is greater. Let's prove it.

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

It remains to prove the last of the listed properties of degrees with natural exponents. Let's formulate it. Of two degrees with natural indicators and the same positive bases, less than one, the greater is the degree, the indicator of which is less; and of two degrees with natural indicators and the same bases, greater than one, the greater is the degree, the indicator of which is greater. We pass to the proof of this property.

Let us prove that for m> n and 0 0 by virtue of the initial condition m> n, whence it follows that for 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 placing a n outside the 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 due to 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, as required. This property is illustrated by the inequality 3 7> 3 2.

Properties of degrees with integer exponents

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

The degree with a negative integer exponent, as well as a degree with a zero exponent, we determined so that all properties of degrees with natural exponents, expressed by equalities, remained true. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the exponents are nonzero.

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

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

For a = 0, the degrees 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 down 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 natural and integer exponents, as well as the properties of actions with real numbers. As an example, let us prove that the property of degree to degree holds for both positive integers and non-positive integers. For this, it is necessary 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.

Now let us prove that (a - p) q = a (- p) q. By definition of a degree with an integer negative exponent, then ... By the property of the quotient in degree, 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 .

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 written 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 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, as required.

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 degrees with rational exponents

We determined a degree with a fractional exponent by extending the properties of a degree with a whole exponent to it. In other words, fractional exponents have the same properties as 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. Here are the proofs.

By definition of a 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 a 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 obtained degree can be transformed as follows:. This completes the proof.

The second property of degrees with fractional exponents is proved in exactly the same way:

Other equalities are proved by similar principles:

We pass to the proof of the following 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. The conditions p<0 и p>0 in this case, 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 us prove that for rational numbers p and q, p> q for 0 0 - inequality a p> a q. We can always bring the 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 natural. 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 degrees 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 accordingly as and ... And the definition of the degree with a rational exponent allows you to go to inequalities and, respectively. Hence, 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, we can conclude that it has all the properties of degrees with a rational exponent. 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.

Hence, we can conclude that degrees with any real exponents p and q for a> 0 have the same properties.

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