Degree with a negative exponent definition examples. How to raise a number to a negative power - examples with descriptions in Excel

In one of the previous articles we already mentioned the power of a number. Today we will try to navigate the process of finding its meaning. Scientifically speaking, we will figure out how to raise to a power correctly. We will figure out how this process is carried out, and at the same time we will touch on all possible exponents: natural, irrational, rational, integer.

So, let's take a closer look at the solutions to the examples and find out what it means:

Definition of the concept.
Raising to negative art.
A whole indicator.
Raising a number to an irrational power.

Here is a definition that accurately reflects the meaning: “Exponentiation is the definition of the value of a power of a number.”

Accordingly, raising the number a in Art. r and the process of finding the value of the degree a with the exponent r are identical concepts. For example, if the task is to calculate the value of the power (0.6)6″, then it can be simplified to the expression “Raise the number 0.6 to the power of 6.”

After this, you can proceed directly to the construction rules.

Raising to a negative power

For clarity, you should pay attention to the following chain of expressions:

110=0.1=1* 10 minus 1 tbsp.,

1100=0.01=1*10 in minus 2 degrees,

11000=0.0001=1*10 in minus 3 st.,

110000=0.00001=1*10 to minus 4 degrees.

Thanks to these examples, you can clearly see the ability to instantly calculate 10 to any minus power. For this purpose, it is enough to simply shift the decimal component:

10 to the -1 degree - before one there is 1 zero;
in -3 - three zeros before one;
in -9 there are 9 zeros and so on.

It is also easy to understand from this diagram how much 10 minus 5 tbsp will be. -

1100000=0,000001=(1*10)-5.

How to raise a number to a natural power

Remembering the definition, we take into account that the natural number a in Art. n equals the product of n factors, each of which equals a. Let's illustrate: (a*a*…a)n, where n is the number of numbers that are multiplied. Accordingly, in order to raise a to n, it is necessary to calculate the product of the following form: a*a*…a divided by n times.

From this it becomes obvious that raising to natural st. relies on the ability to perform multiplication(this material is covered in the section on multiplying real numbers). Let's look at the problem:

Raise -2 to the 4th st.

We are dealing with a natural indicator. Accordingly, the course of the decision will be as follows: (-2) in Art. 4 = (-2)*(-2)*(-2)*(-2). Now all that remains is to multiply the integers: (-2)*(-2)*(-2)*(-2). We get 16.

Answer to the problem:

(-2) in Art. 4=16.

Example:

Calculate the value: three point two sevenths squared.

This example equals the following product: three point two sevenths multiplied by three point two sevenths. Recalling how mixed numbers are multiplied, we complete the construction:

3 point 2 sevenths multiplied by themselves;
equals 23 sevenths multiplied by 23 sevenths;
equals 529 forty-ninths;
we reduce and we get 10 thirty-nine forty-ninths.

Answer: 10 39/49

Regarding the issue of raising to an irrational exponent, it should be noted that calculations begin to be carried out after the completion of preliminary rounding of the basis of the degree to any digit that would allow obtaining the value with a given accuracy. For example, we need to square the number P (pi).

We start by rounding P to hundredths and get:

P squared = (3.14)2=9.8596. However, if we reduce P to ten thousandths, we get P = 3.14159. Then squaring gives a completely different number: 9.8695877281.

It should be noted here that in many problems there is no need to raise irrational numbers to powers. As a rule, the answer is entered either in the form of the actual degree, for example, the root of 6 to the power of 3, or, if the expression allows, its transformation is carried out: root of 5 to 7 degrees = 125 root of 5.

How to raise a number to an integer power

This algebraic manipulation is appropriate take into account for the following cases:

for integers;
for a zero indicator;
for a positive integer exponent.

Since almost all positive integers coincide with the mass of natural numbers, setting to a positive integer power is the same process as setting in Art. natural. We described this process in the previous paragraph.

Now let's talk about calculating st. null. We have already found out above that the zero power of the number a can be determined for any non-zero a (real), while a in Art. 0 will equal 1.

Accordingly, raising any real number to the zero st. will give one.

For example, 10 in st. 0=1, (-3.65)0=1, and 0 in st. 0 cannot be determined.

In order to complete raising to an integer power, it remains to decide on the options for integers negative values. We remember that Art. from a with an integer exponent -z will be defined as a fraction. The denominator of the fraction is st. with a positive integer value, the value of which we have already learned to find. Now all that remains is to consider an example of construction.

Example:

Calculate the value of the number 2 cubed with a negative integer exponent.

Solution process:

According to the definition of a degree with a negative exponent, we denote: two minus 3 degrees. equals one to two to the third power.

The denominator is calculated simply: two cubed;

3 = 2*2*2=8.

Answer: two to the minus 3rd art. = one eighth.

A number raised to a power They call a number that is multiplied by itself several times.

Power of a number with a negative value (a - n) can be determined in a similar way to how the power of the same number with a positive exponent is determined (a n) . However, it also requires additional definition. The formula is defined as:

a-n = (1/a n)

The properties of negative powers of numbers are similar to powers with a positive exponent. Presented equation a m/a n= a m-n may be fair as

« Nowhere, as in mathematics, does the clarity and accuracy of the conclusion allow a person to wriggle out of an answer by talking around the question».

A. D. Alexandrov

at n more m , and with m more n . Let's look at an example: 7 2 -7 5 =7 2-5 =7 -3 .

First you need to determine the number that acts as a definition of the degree. b=a(-n) . In this example -n is an exponent b - the desired numerical value, a - the base of the degree in the form of a natural numeric value. Then determine the module, that is, the absolute value of a negative number, which acts as an exponent. Calculate the degree of a given number relative to an absolute number, as an indicator. The value of the degree is found by dividing one by the resulting number.

Rice. 1

Consider the power of a number with a negative fractional exponent. Let's imagine that the number a is any positive number, numbers n And m - integers. According to definition a , which is raised to the power - equals one divided by the same number with a positive power (Figure 1). When the power of a number is a fraction, then in such cases only numbers with positive exponents are used.

Worth remembering that zero can never be an exponent of a number (the rule of division by zero).

The spread of such a concept as a number became such manipulations as measurement calculations, as well as the development of mathematics as a science. The introduction of negative values was due to the development of algebra, which gave general solutions arithmetic problems, regardless of their specific meaning and initial numerical data. In India, back in the 6th-11th centuries, negative numbers were systematically used when solving problems and were interpreted in the same way as today. In European science, negative numbers began to be widely used thanks to R. Descartes, who gave a geometric interpretation of negative numbers as the directions of segments. It was Descartes who proposed the designation of a number raised to a power to be displayed as a two-story formula a n .

The power is used to simplify the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation for this transition is given in the first section of this article). Degrees make it easier to write long or complex expressions or equations; powers are also easy to add and subtract, resulting in a simplified expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).

Note: if you need to solve an exponential equation (in such an equation the unknown is in the exponent), read.

Steps

Solving simple problems with degrees

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a power problem by hand, rewrite the power as a multiplication operation, where the base of the power is multiplied by itself. For example, given a degree 3 4 (\displaystyle 3^(4)). In this case, the base of power 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:

First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two fours and then replace them with the result. Like this:

4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=4*4*4*4*4)
- 4 ∗ 4 = 16 (\displaystyle 4*4=16)

Multiply the result (16 in our example) by the next number. Each next result will increase proportionally. In our example, multiply 16 by 4. Like this:
- 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
  - 16 ∗ 4 = 64 (\displaystyle 16*4=64)
- 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
  - 64 ∗ 4 = 256 (\displaystyle 64*4=256)
- 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
  - 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
- Continue multiplying the result of the first two numbers by the next number until you get your final answer. To do this, multiply the first two numbers, and then multiply the resulting result by the next number in the sequence. This method is valid for any degree. In our example you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .
Solve the following problems. Check your answer using a calculator.
- 8 2 (\displaystyle 8^(2))
- 3 4 (\displaystyle 3^(4))
- 10 7 (\displaystyle 10^(7))
On your calculator, look for the key labeled "exp" or " x n (\displaystyle x^(n))", or "^". Using this key you will raise a number to a power. It is almost impossible to calculate a degree with a large indicator manually (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click “View” -> “Engineering”. To switch to normal mode, click “View” -> “Normal”.
- Check your answer using search engine(Google or Yandex). Using the "^" key on your computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for you to study).
Addition, subtraction, multiplication of powers
1. You can add and subtract degrees only if they have the same bases. If you need to add powers with the same bases and exponents, then you can replace the addition operation with the multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented in the form 1 ∗ 4 5 (\displaystyle 1*4^(5)); Thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply that degree and this number. In our example, raise 4 to the fifth power, and then multiply the resulting result by 2. Remember that the addition operation can be replaced by the multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:
  - 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
  - 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
  - 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
  - 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))
2. When multiplying powers with the same base, their exponents are added (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:
  
  When raising a power to a power, the exponents are multiplied. For example, a degree is given. Since exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The point of this rule is that you multiply by powers (x 2) (\displaystyle (x^(2))) on itself five times. Like this:
  - (x 2) 5 (\displaystyle (x^(2))^(5))
  - (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
  - Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))
3. A power with a negative exponent should be converted to a fraction (reverse power). It doesn't matter if you don't know what a reciprocal degree is. If you are given a degree with a negative exponent, e.g. 3 − 2 (\displaystyle 3^(-2)), write this degree in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:
  
  When dividing degrees with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .
  - The power in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.
4. Below are some expressions that will help you learn to solve problems with exponents. The expressions given cover the material presented in this section. To see the answer, simply select the empty space after the equals sign.
Solving problems with fractional exponents
1. If the exponent is an improper fraction, then the exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule of multiplying powers. For example, a degree is given. Convert such a power into a root whose power is equal to the denominator of the fractional exponent, and then raise this root to a power equal to the numerator of the fractional exponent. To do this, remember that 5 3 (\displaystyle (\frac (5)(3))) = (1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5). In our example:
  - x 5 3 (\displaystyle x^(\frac (5)(3)))
  - x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x)))
  - x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3))) = (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
2. Some calculators have a button to calculate exponents (you must first enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
3. Remember that any number to the first power is equal to itself, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, e.g. 5 ∗ 1 = 5 (\displaystyle 5*1=5) And 5 / 1 = 5 (\displaystyle 5/1=5).
4. Know that the power 0 0 does not exist (such a power has no solution). If you try to solve such a degree on a calculator or on a computer, you will receive an error. But remember that any number to the zero power is 1, for example, 4 0 = 1. (\displaystyle 4^(0)=1.)
5. In higher mathematics, which operates with imaginary numbers: e a i x = c o s a x + i s i n a x (\displaystyle e^(a)ix=cosax+isinax), Where i = (− 1) (\displaystyle i=(\sqrt (())-1)); e is a constant approximately equal to 2.7; a is an arbitrary constant. The proof of this equality can be found in any textbook on higher mathematics.
- As the exponent increases, its value increases greatly. So if the answer seems wrong to you, it may actually be correct. You can test this by plotting any exponential function, such as 2 x.

It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds equal powers of identical variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is equal to 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3.

It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Multiplying powers

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.

Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n;

And a m is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents of the powers.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y -n .y -m = y -n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers equal to the sum or the difference of their squares.

If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.

So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.

Division of degrees

Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.

Thus, a 3 b 2 divided by b 2 is equal to a 3.

Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$

Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing degrees with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1. That is, $\frac(yyy)(yy) = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.

Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3

The rule is also true for numbers with negative values of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$

It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce the exponents by $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Decrease the exponents by $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.

3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.

In this material we will look at what a power of a number is. In addition to the basic definitions, we will formulate what powers with natural, integer, rational and irrational exponents are. As always, all concepts will be illustrated with example problems.

Yandex.RTB R-A-339285-1

First, let's formulate the basic definition of a degree with a natural exponent. To do this, we need to remember the basic rules of multiplication. Let us clarify in advance that for now we will take a real number as a base (denoted by the letter a), and a natural number as an indicator (denoted by the letter n).

Definition 1

The power of a number a with natural exponent n is the product of the nth number of factors, each of which is equal to the number a. The degree is written like this: a n, and in the form of a formula its composition can be represented as follows:

For example, if the exponent is 1 and the base is a, then the first power of a is written as a 1. Given that a is the value of the factor and 1 is the number of factors, we can conclude that a 1 = a.

In general, we can say that a degree is a convenient form of recording large quantity equal factors. So, a record of the form 8 8 8 8 can be shortened to 8 4 . In much the same way, the product helps us avoid writing a large number of terms (8 + 8 + 8 + 8 = 8 · 4); We have already discussed this in the article devoted to the multiplication of natural numbers.

How to correctly read the degree entry? The generally accepted option is “a to the power of n”. Or you can say “nth power of a” or “anth power”. If, say, in the example we encountered the entry 8 12 , we can read "8 to the 12th power", "8 to the power of 12" or "12th power of 8".

The second and third powers of numbers have their own established names: square and cube. If we see the second power, for example, the number 7 (7 2), then we can say “7 squared” or “square of the number 7”. Similarly, the third degree is read like this: 5 3 - this is the “cube of the number 5” or “5 cubed.” However, you can also use the standard formulation “to the second/third power”; this will not be a mistake.

Example 1

Let's look at an example of a degree with a natural exponent: for 5 7 five will be the base, and seven will be the exponent.

The base does not have to be an integer: for the degree (4 , 32) 9 the base will be the fraction 4, 32, and the exponent will be nine. Pay attention to the parentheses: this notation is made for all powers whose bases differ from natural numbers.

For example: 1 2 3, (- 3) 12, - 2 3 5 2, 2, 4 35 5, 7 3.

What are parentheses for? They help avoid errors in calculations. Let's say we have two entries: (− 2) 3 And − 2 3 . The first of these means a negative number minus two raised to a power with a natural exponent of three; the second is the number corresponding to the opposite value of the degree 2 3 .

Sometimes in books you can find a slightly different spelling of the power of a number - a^n(where a is the base and n is the exponent). That is, 4^9 is the same as 4 9 . If n is a multi-digit number, it is placed in parentheses. For example, 15 ^ (21) , (− 3 , 1) ^ (156) . But we will use the notation a n as more common.

It’s easy to guess how to calculate the value of an exponent with a natural exponent from its definition: you just need to multiply a nth number of times. We wrote more about this in another article.

The concept of degree is the inverse of another mathematical concept - the root of a number. If we know the value of the power and the exponent, we can calculate its base. The degree has some specific properties that are useful for solving problems, which we discussed in a separate material.

Exponents can include not only natural numbers, but also any integer values in general, including negative ones and zeros, because they also belong to the set of integers.

Definition 2

The power of a number with a positive integer exponent can be represented as a formula: .

In this case, n is any positive integer.

Let's understand the concept of zero degree. To do this, we use an approach that takes into account the quotient property for powers with equal bases. It is formulated like this:

Definition 3

Equality a m: a n = a m − n will be true under the following conditions: m and n are natural numbers, m< n , a ≠ 0 .

The last condition is important because it avoids division by zero. If the values of m and n are equal, then we get the following result: a n: a n = a n − n = a 0

But at the same time a n: a n = 1 is a quotient equal numbers a n and a. It turns out that the zero power of any non-zero number is equal to one.

However, such a proof does not apply to zero to the zeroth power. To do this, we need another property of powers - the property of products of powers with equal bases. It looks like this: a m · a n = a m + n .

If n is equal to 0, then a m · a 0 = a m(this equality also proves to us that a 0 = 1). But if and is also equal to zero, our equality takes the form 0 m · 0 0 = 0 m, It will be true for any natural value of n, and it does not matter what exactly the value of the degree is equal to 0 0 , that is, it can be equal to any number, and this will not affect the accuracy of the equality. Therefore, a notation of the form 0 0 does not have its own special meaning, and we will not attribute it to it.

If desired, it is easy to check that a 0 = 1 converges with the degree property (a m) n = a m n provided that the base of the degree is not zero. Thus, the power of any non-zero number with exponent zero is one.

Example 2

Let's look at an example with specific numbers: So, 5 0 - unit, (33 , 3) 0 = 1 , - 4 5 9 0 = 1 , and the value 0 0 undefined.

After the zero degree, we just have to figure out what a negative degree is. To do this, we need the same property of the product of powers with equal bases that we already used above: a m · a n = a m + n.

Let us introduce the condition: m = − n, then a should not be equal to zero. It follows that a − n · a n = a − n + n = a 0 = 1. It turns out that a n and a−n we have mutually reciprocal numbers.

As a result, a to the negative whole power is nothing more than the fraction 1 a n.

This formulation confirms that for a degree with an integer negative exponent, all the same properties are valid that a degree with a natural exponent has (provided that the base is not equal to zero).

Example 3

A power a with a negative integer exponent n can be represented as a fraction 1 a n . Thus, a - n = 1 a n subject to a ≠ 0 and n is any natural number.

Let us illustrate our idea with specific examples:

Example 4

3 - 2 = 1 3 2 , (- 4 . 2) - 5 = 1 (- 4 . 2) 5 , 11 37 - 1 = 1 11 37 1

In the last part of the paragraph, we will try to depict everything that has been said clearly in one formula:

Definition 4

The power of a number with a natural exponent z is: a z = a z, e with l and z - positive integer 1, z = 0 and a ≠ 0, (for z = 0 and a = 0 the result is 0 0, the values of the expression 0 0 are not is defined) 1 a z, if and z is a negative integer and a ≠ 0 ( if z is a negative integer and a = 0 you get 0 z, egoz the value is undetermined)

What are powers with a rational exponent?

We examined cases when the exponent contains an integer. However, you can raise a number to a power even when its exponent contains a fractional number. This is called a power with a rational exponent. In this section we will prove that it has the same properties as other powers.

What are rational numbers? Their set includes both whole and fractional numbers, and fractional numbers can be represented as ordinary fractions (both positive and negative). Let us formulate the definition of the power of a number a with a fractional exponent m / n, where n is a natural number and m is an integer.

We have some degree with a fractional exponent a m n . In order for the power to power property to hold, the equality a m n n = a m n · n = a m must be true.

Given the definition of the nth root and that a m n n = a m, we can accept the condition a m n = a m n if a m n makes sense for the given values of m, n and a.

The above properties of a degree with an integer exponent will be true under the condition a m n = a m n .

The main conclusion from our reasoning is this: the power of a certain number a with a fractional exponent m / n is the nth root of the number a to the power m. This is true if, for given values of m, n and a, the expression a m n remains meaningful.

1. We can limit the value of the base of the degree: let's take a, which for positive values of m will be greater than or equal to 0, and for negative values - strictly less (since for m ≤ 0 we get 0 m, but such a degree is not defined). In this case, the definition of a degree with a fractional exponent will look like this:

A power with a fractional exponent m/n for some positive number a is the nth root of a raised to the power m. This can be expressed as a formula:

For a power with a zero base, this provision is also suitable, but only if its exponent is a positive number.

A power with a base zero and a fractional positive exponent m/n can be expressed as

0 m n = 0 m n = 0 provided m is a positive integer and n is a natural number.

For a negative ratio m n< 0 степень не определяется, т.е. такая запись смысла не имеет.

Let's note one point. Since we introduced the condition that a is greater than or equal to zero, we ended up discarding some cases.

The expression a m n sometimes still makes sense for some negative values of a and some m. Thus, the correct entries are (- 5) 2 3, (- 1, 2) 5 7, - 1 2 - 8 4, in which the base is negative.

2. The second approach is to consider separately the root a m n with even and odd exponents. Then we will need to introduce one more condition: the degree a, in the exponent of which there is a reducible ordinary fraction, is considered to be the degree a, in the exponent of which there is the corresponding irreducible fraction. Later we will explain why we need this condition and why it is so important. Thus, if we have the notation a m · k n · k , then we can reduce it to a m n and simplify the calculations.

If n is an odd number and the value of m is positive and a is any non-negative number, then a m n makes sense. The condition for a to be non-negative is necessary because a root of an even degree cannot be extracted from a negative number. If the value of m is positive, then a can be both negative and zero, because The odd root can be taken from any real number.

Let's combine all the above definitions in one entry:

Here m/n means an irreducible fraction, m is any integer, and n is any natural number.

Definition 5

For any ordinary reducible fraction m · k n · k the degree can be replaced by a m n .

The power of a number a with an irreducible fractional exponent m / n – can be expressed as a m n in the following cases: - for any real a, positive integer values m and odd natural values n. Example: 2 5 3 = 2 5 3, (- 5, 1) 2 7 = (- 5, 1) - 2 7, 0 5 19 = 0 5 19.

For any non-zero real a, negative integer values of m and odd values of n, for example, 2 - 5 3 = 2 - 5 3, (- 5, 1) - 2 7 = (- 5, 1) - 2 7

For any non-negative a, positive integer m and even n, for example, 2 1 4 = 2 1 4, (5, 1) 3 2 = (5, 1) 3, 0 7 18 = 0 7 18.

For any positive a, negative integer m and even n, for example, 2 - 1 4 = 2 - 1 4, (5, 1) - 3 2 = (5, 1) - 3, .

In the case of other values, the degree with a fractional exponent is not determined. Examples of such degrees: - 2 11 6, - 2 1 2 3 2, 0 - 2 5.

Now let’s explain the importance of the condition discussed above: why replace a fraction with a reducible exponent with a fraction with an irreducible exponent. If we had not done this, we would have had the following situations, say, 6/10 = 3/5. Then it should be true (- 1) 6 10 = - 1 3 5 , but - 1 6 10 = (- 1) 6 10 = 1 10 = 1 10 10 = 1 , and (- 1) 3 5 = (- 1) 3 5 = - 1 5 = - 1 5 5 = - 1 .

The definition of a degree with a fractional exponent, which we presented first, is more convenient to use in practice than the second, so we will continue to use it.

Definition 6

Thus, the power of a positive number a with a fractional exponent m/n is defined as 0 m n = 0 m n = 0. In case of negative a the notation a m n does not make sense. Power of zero for positive fractional exponents m/n is defined as 0 m n = 0 m n = 0 , for negative fractional exponents we do not define the degree of zero.

In conclusions, we note that any fractional indicator can be written both in the form of a mixed number and in the form decimal: 5 1 , 7 , 3 2 5 - 2 3 7 .

When calculating, it is better to replace the exponent ordinary fraction and continue to use the definition of degree with a fractional exponent. For the examples above we get:

5 1 , 7 = 5 17 10 = 5 7 10 3 2 5 - 2 3 7 = 3 2 5 - 17 7 = 3 2 5 - 17 7

What are powers with irrational and real exponents?

What are real numbers? Their set includes both rational and irrational numbers. Therefore, in order to understand what a degree with a real exponent is, we need to define degrees with rational and irrational exponents. We have already mentioned rational ones above. Let's deal with irrational indicators step by step.

Example 5

Let's assume that we have an irrational number a and a sequence of its decimal approximations a 0 , a 1 , a 2 , . . . . For example, let's take the value a = 1.67175331. . . , Then

a 0 = 1, 6, a 1 = 1, 67, a 2 = 1, 671, . . . , a 0 = 1.67, a 1 = 1.6717, a 2 = 1.671753, . . .

We can associate sequences of approximations with a sequence of degrees a a 0 , a a 1 , a a 2 , . . . . If we remember what we said earlier about raising numbers to rational powers, then we can calculate the values of these powers ourselves.

Let's take for example a = 3, then a a 0 = 3 1, 67, a a 1 = 3 1, 6717, a a 2 = 3 1, 671753, . . . etc.

The sequence of powers can be reduced to a number, which will be the value of the power with base a and irrational exponent a. As a result: a degree with an irrational exponent of the form 3 1, 67175331. . can be reduced to the number 6, 27.

Definition 7

The power of a positive number a with an irrational exponent a is written as a a . Its value is the limit of the sequence a a 0 , a a 1 , a a 2 , . . . , where a 0 , a 1 , a 2 , . . . are successive decimal approximations of the irrational number a. A degree with a zero base can also be defined for positive irrational exponents, with 0 a = 0 So, 0 6 = 0, 0 21 3 3 = 0. But this cannot be done for negative ones, since, for example, the value 0 - 5, 0 - 2 π is not defined. A unit raised to any irrational power remains a unit, for example, and 1 2, 1 5 in 2 and 1 - 5 will be equal to 1.

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