Degree of number: definitions, designation, examples. Exponential function

Degree with a rational exponent, its properties.

Expression a n defined for all a and n, except for the case a=0 for n≤0. Recall the properties of such powers.

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 \u003d a m-n (a ≠ 0); (a m) n = a mn ; (ab) n = a n * b n ; (b≠0); a 1 =a; a 0 =1 (a≠0).

Note also the following property:

If m>n, then a m > a n for a> 1 and a m<а n при 0<а<1.

In this subsection, we generalize the notion of a power of a number by giving meaning to expressions like 2 0.3 , 8 5/7 , 4 -1/2 etc. It is natural to give a definition in such a way that the powers with rational exponents have the same properties (or at least part of them) as the powers with an integer exponent. Then, in particular, the nth power of the numbershould be equal to a m . Indeed, if the property

(a p) q =a pq

is performed, then

The last equality means (by definition of the nth root) that the numbermust be the nth root of a m .

Definition.

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

So by definition

(1)

The power of 0 is only defined for positive exponents; by definition 0 r = 0 for any r>0.

Degree with an irrational exponent.

irrational numbercan be represented aslimit of a sequence of rational numbers: .

Let . Then there are powers with rational exponent . It can be proved that the sequence of these powers is convergent. The limit of this sequence is called degree with base and irrational exponent: .

Let's plot several points of the graph of the function y \u003d 2 x having previously calculated with the help of a calculator the values ​​2 x on the interval [-2; 3] with a step of 1/4 (Fig. 1, a), and then with a step of 1/8 (Fig. 1, b). Continuing mentally the same constructions with a step of 1/16, 1/32, etc., we see that the resulting points can be connected by a smooth curve, which it is natural to consider the graph of some function, already defined and increasing on the entire number line and taking the valuesat rational points(Fig. 1, c). Having constructed a sufficiently large number of points of the graph of the function, you can make sure that this function has similar properties (the difference is that the function decreases on R).

These observations suggest that it is possible to define the numbers 2α and for every irrational α such that the functions given by the formulas y=2 x and will be continuous, and the function y \u003d 2 x increases, and the functiondecreases along the whole number line.

Let us describe in general terms how the number a α for irrational α for a>1. We want to ensure that the function y = a x was increasing. Then for any rational r 1 and r 2 such that r 1<αmust satisfy the inequalities a r1<а α <а r 1 .

Choosing values ​​r 1 and r2 approaching x, it can be seen that the corresponding values ​​of a r 1 and a r 2 will differ little. It can be proved that there exists, and moreover, only one, a number y that is greater than all a r1 for all rational r 1 and less than all a r 2 for all rational r 2 . This number y is, by definition, a α .

For example, having calculated with the help of a calculator the values ​​2 x at points x n and x` n , where x n and x` n - decimal approximations of a numberwe find that the closer x n and x` n to , the less they differ 2 x n and 2 x` n .

Since then

and, therefore,

Similarly, considering the following decimal approximationsby deficiency and excess, we arrive at the relations

;

;

;

;

.

Meaning calculated on the calculator is:

.

The number a α for 0<α<1. Кроме того полагают 1 α =1 for any α and 0α =0 for α>0.

Exponential function.

At a > 0, a = 1, function defined y=a x, which is different from the constant. This function is called exponential function with basea.

y= a x at a> 1:

Graphs of exponential functions with base 0< a < 1 и a> 1 are shown in the figure.

Basic properties of the exponential function y= a x at 0< a < 1:

• The scope of the function is the entire number line.
• Function range - span (0; + ) .
• The function is strictly monotonically increasing on the whole number line, that is, if x 1 < x 2 , then a x 1 > a x 2 .
• At x= 0 function value is 1.
• If a x> 0 , then 0< a < 1 and if x < 0, то a x > 1.
To the general properties of the exponential function as at0< a < 1, так и при a > 1 are:
• a x 1 a x 2 = a x 1 + x 2 , for everyone x 1 and x 2.
• a − x= ( a x) − 1 = 1 ax for anyone x.
• na x= a

Degree with a rational exponent, its properties.

Expression a n defined for all a and n, except for the case a=0 for n≤0. Recall the properties of such powers.

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 \u003d a m-n (a ≠ 0); (a m) n = a mn ; (ab) n = a n * b n ; (b≠0); a 1 =a; a 0 =1 (a≠0).

Note also the following property:

If m>n, then a m > a n for a> 1 and a m<а n при 0<а<1.

In this subsection, we generalize the notion of a power of a number by giving meaning to expressions like 2 0.3 , 8 5/7 , 4 -1/2 etc. It is natural to give a definition in such a way that the powers with rational exponents have the same properties (or at least part of them) as the powers with an integer exponent. Then, in particular, the nth power of the numbershould be equal to a m . Indeed, if the property

(a p) q =a pq

is performed, then

The last equality means (by definition of the nth root) that the numbermust be the nth root of a m .

Definition.

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

So by definition

(1)

The power of 0 is only defined for positive exponents; by definition 0 r = 0 for any r>0.

Degree with an irrational exponent.

irrational numbercan be represented aslimit of a sequence of rational numbers: .

Let . Then there are powers with rational exponent . It can be proved that the sequence of these powers is convergent. The limit of this sequence is called degree with base and irrational exponent: .

Let's plot several points of the graph of the function y \u003d 2 x having previously calculated with the help of a calculator the values ​​2 x on the interval [-2; 3] with a step of 1/4 (Fig. 1, a), and then with a step of 1/8 (Fig. 1, b). Continuing mentally the same constructions with a step of 1/16, 1/32, etc., we see that the resulting points can be connected by a smooth curve, which it is natural to consider the graph of some function, already defined and increasing on the entire number line and taking the valuesat rational points(Fig. 1, c). Having constructed a sufficiently large number of points of the graph of the function, you can make sure that this function has similar properties (the difference is that the function decreases on R).

These observations suggest that it is possible to define the numbers 2α and for every irrational α such that the functions given by the formulas y=2 x and will be continuous, and the function y \u003d 2 x increases, and the functiondecreases along the whole number line.

Let us describe in general terms how the number a α for irrational α for a>1. We want to ensure that the function y = a x was increasing. Then for any rational r 1 and r 2 such that r 1<αmust satisfy the inequalities a r1<а α <а r 1 .

Choosing values ​​r 1 and r2 approaching x, it can be seen that the corresponding values ​​of a r 1 and a r 2 will differ little. It can be proved that there exists, and moreover, only one, a number y that is greater than all a r1 for all rational r 1 and less than all a r 2 for all rational r 2 . This number y is, by definition, a α .

For example, having calculated with the help of a calculator the values ​​2 x at points x n and x` n , where x n and x` n - decimal approximations of a numberwe find that the closer x n and x` n to , the less they differ 2 x n and 2 x` n .

Since then

and, therefore,

Similarly, considering the following decimal approximationsby deficiency and excess, we arrive at the relations

;

;

;

;

.

Meaning calculated on the calculator is:

.

The number a α for 0<α<1. Кроме того полагают 1 α =1 for any α and 0α =0 for α>0.

Exponential function.

At a > 0, a = 1, function defined y=a x, which is different from the constant. This function is called exponential function with basea.

y= a x at a> 1:

Graphs of exponential functions with base 0< a < 1 и a> 1 are shown in the figure.

Basic properties of the exponential function y= a x at 0< a < 1:

• The scope of the function is the entire number line.
• Function range - span (0; + ) .
• The function is strictly monotonically increasing on the whole number line, that is, if x 1 < x 2 , then a x 1 > a x 2 .
• At x= 0 function value is 1.
• If a x> 0 , then 0< a < 1 and if x < 0, то a x > 1.
To the general properties of the exponential function as at0< a < 1, так и при a > 1 are:
• a x 1 a x 2 = a x 1 + x 2 , for everyone x 1 and x 2.
• a − x= ( a x) − 1 = 1 ax for anyone x.
• na x= a

PART II. CHAPTER 6
SEQUENCES OF NUMBERS

The concept of degree with an irrational exponent

Let a be some positive number and a be irrational.
What meaning should be given to the expression a*?
To make the presentation more illustrative, we will carry it out on a particular
example. Namely, let's put a - 2 and a = 1, 624121121112. . . .
Here, a is an infinite decimal fraction composed according to such
law: starting from the fourth decimal place, for the image a
only numbers 1 and 2 are used, and at the same time the number of digits 1,
written in a row before the number 2, all the time increases by
one. The fraction a is non-periodic, since otherwise the number of digits is 1,
recorded in a row in his image would be limited.
Therefore, a is an irrational number.
So what is the meaning of the expression
21, in2Sh1Sh1Sh11Sh11Sh. . . R
To answer this question, we compose sequences of values
and with deficiency and excess up to (0.1)*. Get
1,6; 1,62; 1,624; 1,6241; …, (1)
1,7; 1,63; 1,625; 1,6242; . . . (2)
We compose the corresponding sequences of powers of the number 2:
2M. 2M*; 21*624; 21'62*1; …, (3)
21D. 21"63; 2*»62Vu 21.6Sh; . (four)
Sequence (3) is increasing because the sequence
(1) (Theorem 2 § 6).
The sequence (4) is decreasing because the sequence
(2).
Each member of the sequence (3) is less than each member of the sequence
(4), and thus the sequence (3) is bounded
from above, and the sequence (4) is bounded from below.
Based on the monotone bounded sequence theorem
each of the sequences (3) and (4) has a limit. If a

384 The concept of degree with an irrational exponent . .

now, it turns out that the difference of sequences (4) and (3) converges
to zero, then it will follow that both of these sequences,
have a common limit.
Difference of the first terms of sequences (3) and (4)
21-7 - 21’* = 2|, in (20*1 - 1)< 4 (У 2 - 1).
Difference of the second terms
21'63 - 21.62 = 21.62 (2°'01 - 1)< 4 (l0 j/2f - 1) и т. д.
Difference of n-th terms
0,0000. ..0 1
2>.««…(2 "- 1)< 4 (l0“/ 2 - 1).
Based on Theorem 3 § 6
lim 10″ / 2 = 1.
So sequences (3) and (4) have a common limit. This
limit is the only real number that is greater than
all members of the sequence (3) and less than all members of the sequence
(4), and it is expedient to consider it as the exact value of 2*.
From what has been said, it follows that it is generally advisable to take
the following definition:
Definition. If a > 1, then the power of a with irrational
exponent a is such a real number,
which is greater than all the powers of this number whose exponents are
rational approximations a with a disadvantage, and less than all powers
this number, whose exponents are rational approximations a c
excess.
If a<^ 1, то степенью числа а с иррациональным показателем а
a real number is called which is greater than all powers
this number, whose exponents are rational approximations a
in excess, and less than all the powers of this number, the indicators of which
are rational approximations a with a drawback.
.If a - 1, then its degree with an irrational exponent a
is 1.
Using the concept of a limit, this definition can be formulated
So:
Power of a positive number with an irrational exponent
and is called the limit to which the sequence tends
rational powers of this number, provided that the sequence
indicators of these degrees tends to a, i.e.
aa = lim ah
b - *
13 D, K. Fatshcheev, I. S. Sominsky

Information boom In biology - colonies of microbes in a petri dish Rabbits in Australia Chain reactions - in chemistry In physics - radioactive decay, change in atmospheric pressure with a change in altitude, cooling of the body. In physics - radioactive decay, change in atmospheric pressure with a change in height, cooling of the body. The release of adrenaline into the blood and its destruction They also claim that the amount of information doubles every 10 years. They also claim that the amount of information doubles every 10 years.

(3/5) -1 a 1 3 1/2 (4/9) 0 a *81 (1/2) -3 a -n 36 1/2* 8 1/ /3 2 -3.5

Expression 2 x 2 2 = 4 2 5 = = = 1/2 4 = 1/16 2 4/3 = 32 4 = .5 = 1/2 3.5 = 1/2 7= 1/(8 2) = 2/16 2)=

3=1, … 1; 1.7 1.73; 1.732;1.73205; 1, ;… the sequence increases 2 1 ; 2 1.7; 2 1.73; 2 1.732; 2 1.73205; 2 1, ;… the sequence increases Limited, which means it converges to one limit - value 2 3

One can define π 0

10 10 18 Function properties y = a x n \ n a >10 10 10 10 10 title="(!LANG: Function properties y = a x n \ n a >10 21

The amount of information doubles every 10 years On the Ox axis - according to the law of arithmetic progression: 1,2,3,4…. On the Oy axis - according to the law of geometric progression: 2 1.2 2.2 3.2 4 ... The graph of the exponential function, it is called the exponent (from the Latin exponere - to flaunt)

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 definition of a power with a natural exponent, the power 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. quotient property in kind (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 natural numbers such that m>n , then at 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 degrees 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 .

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 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 of a m−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 proved in the previous paragraph.

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

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

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 (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 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 recorded properties, 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 us prove that for rational numbers p and q , p>q for 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.

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