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

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

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

Definition.

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

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

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

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

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

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

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

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

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

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

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

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

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

Definition.

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

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

Definition.

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

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

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

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

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

Definition.

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



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

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

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

a-n = (1 / a n)

The properties of negative values ​​of powers of numbers are similar to powers with a positive exponent. Represented Equation a m / a n = a m-n can be fair as

« Nowhere, as in mathematics, the clarity and accuracy of the conclusion does not allow a person to get away from the answer by talking around the question.».

A. D. Alexandrov

at n more m , as well as 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 indicator of the degree b - desired numerical value, a - the base of the degree as a natural numerical value. Then determine the module, that is, the absolute value of a negative number, which acts as an exponent. Calculate the degree of the given number relative to the absolute number as an indicator. The value of the degree is found by dividing one by the resulting number.

Rice. one

Consider the power of a number with a negative fractional exponent. Imagine that the number a is any positive number, the numbers n and m - integers. By definition a , which is raised to the power - equals one divided by the same number with a positive degree (Fig. 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 began 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 to arithmetic problems, regardless of their specific meaning and initial numerical data. In India, back in the 6th-11th centuries, negative values ​​of 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 directions of segments. It was Descartes who suggested that the number raised to a power be displayed as a two-story formula a n .

Obviously, numbers with powers can be added like other quantities , by adding them one by one 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 the same powers of the same variables can be added or subtracted.

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

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

But degrees various variables and various degrees identical variables, must be added by adding them to 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 neither twice the square of a, but 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 subtrahend 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

Power multiplication

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

So, 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 the same 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 sum 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 is;

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.

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 is equal to the sum or difference of their squares.

If 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 degree.

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 divisor, or by placing them in the form of a fraction.

So a 3 b 2 divided by b 2 is 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 powers 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:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
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 the 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 in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Reduce the exponents in $\frac(6x^6)(3x^5)$. 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 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.

The exponent is used to make it easier to write the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation of such a transition is given in the first section of this article). Powers make it easier to write long or complex expressions or equations; also, powers are easily added and subtracted, resulting in a simplification of an 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 powers

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a problem with exponents manually, rewrite the exponent as a multiplication operation, where the base of the exponent is multiplied by itself. For example, given the degree 3 4 (\displaystyle 3^(4)). In this case, the base of degree 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 quadruples, 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)

1. Multiply the result (16 in our example) by the next number. Each subsequent 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)
   • Keep multiplying the result of multiplying the first two numbers by the next number until you get the final answer. To do this, multiply the first two numbers, and then multiply the 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) .

2. Solve the following problems. Check your answer with a calculator.

   • 8 2 (\displaystyle 8^(2))
   • 3 4 (\displaystyle 3^(4))
   • 10 7 (\displaystyle 10^(7))

3. On the calculator, look for the key labeled "exp", or " x n (\displaystyle x^(n))", or "^". With this key you will raise a number to a power. It is practically impossible to manually calculate the degree with a large exponent (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" -\u003e "Engineering". To switch to normal mode, click "View" -\u003e "Normal".

   • Check the received answer using a search engine (Google or Yandex). Using the "^" key on the computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for study).

   Addition, subtraction, multiplication of powers

   1. You can add and subtract powers only if they have the same base. If you need to add powers with the same bases and exponents, then you can replace the addition operation with a 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 as 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 such a degree and this number. In our example, raise 4 to the fifth power, and then multiply the result by 2. Remember that the addition operation can be replaced by a 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. In this way, 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, given a degree. Since the exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The meaning of this rule is that you multiply the power (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. An exponent with a negative exponent should be converted to a fraction (to the inverse power). It doesn't matter if you don't know what a reciprocal is. If you are given a degree with a negative exponent, for example, 3 − 2 (\displaystyle 3^(-2)), write this power 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 powers 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). In this way, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .

      • The degree 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 to help you learn how to solve power problems. The above expressions cover the material presented in this section. To see the answer, just highlight the empty space after the equals sign.

   Solving problems with fractional exponents

   A degree with a fractional exponent (for example, ) is converted to a root extraction operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x(\displaystyle(\sqrt(x))). It does not matter what number is in the denominator of the fractional exponent. For example, x 1 4 (\displaystyle x^(\frac (1)(4))) is the fourth root of "x" x 4 (\displaystyle (\sqrt[(4)](x))) .

   1. If the exponent is an improper fraction, then such an exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule for multiplying powers. For example, given a degree. Turn that exponent into a root whose exponent is equal to the denominator of the fractional exponent, and then raise that root to the exponent 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 for calculating exponents (first you need to enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
   3. Remember that any number is equal to itself to the first power, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, for example, 5 ∗ 1 = 5 (\displaystyle 5*1=5) and 5 / 1 = 5 (\displaystyle 5/1=5).
   4. Know that the degree 0 0 does not exist (such a degree has no solution). When you try to solve such a degree on a calculator or on a computer, you will get an error. But remember that any number to the power of zero is equal to 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.

   Warnings

   • As the exponent increases, its value greatly increases. Therefore, if the answer seems wrong to you, in fact it may turn out to be true. You can check this by plotting any exponential function, such as 2 x .

Raising to a negative power is one of the basic elements of mathematics, which is often encountered in solving algebraic problems. Below is a detailed instruction.

How to raise to a negative power - theory

When we take a number to the usual power, we multiply its value several times. For example, 3 3 \u003d 3 × 3 × 3 \u003d 27. With a negative fraction, the opposite is true. The general form according to the formula will be as follows: a -n = 1/a n . Thus, to raise a number to a negative power, you need to divide the unit by the given number, but already to a positive power.

How to raise to a negative power - examples on ordinary numbers

With the above rule in mind, let's solve a few examples.

4 -2 = 1/4 2 = 1/16
Answer: 4 -2 = 1/16

4 -2 = 1/-4 2 = 1/16.
The answer is -4 -2 = 1/16.

But why is the answer in the first and second examples the same? The fact is that when a negative number is raised to an even power (2, 4, 6, etc.), the sign becomes positive. If the degree were even, then the minus is preserved:

4 -3 = 1/(-4) 3 = 1/(-64)

How to raise to a negative power - numbers from 0 to 1

Recall that when a number between 0 and 1 is raised to a positive power, the value decreases as the power increases. So for example, 0.5 2 = 0.25. 0.25

Example 3: Calculate 0.5 -2
Solution: 0.5 -2 = 1/1/2 -2 = 1/1/4 = 1×4/1 = 4.
Answer: 0.5 -2 = 4

Parsing (sequence of actions):

• Convert decimal 0.5 to fractional 1/2. It's easier.
  Raise 1/2 to a negative power. 1/(2) -2 . Divide 1 by 1/(2) 2 , we get 1/(1/2) 2 => 1/1/4 = 4

Example 4: Calculate 0.5 -3
Solution: 0.5 -3 = (1/2) -3 = 1/(1/2) 3 = 1/(1/8) = 8

Example 5: Calculate -0.5 -3
Solution: -0.5 -3 = (-1/2) -3 = 1/(-1/2) 3 = 1/(-1/8) = -8
Answer: -0.5 -3 = -8

Based on the 4th and 5th examples, we will draw several conclusions:

• For a positive number in the range from 0 to 1 (example 4), raised to a negative power, the even or odd degree is not important, the value of the expression will be positive. In this case, the greater the degree, the greater the value.
• For a negative number between 0 and 1 (example 5), raised to a negative power, the even or odd degree is unimportant, the value of the expression will be negative. In this case, the higher the degree, the lower the value.

How to raise to a negative power - the power as a fractional number

Expressions of this type have the following form: a -m/n , where a is an ordinary number, m is the numerator of the degree, n is the denominator of the degree.

Consider an example:
Calculate: 8 -1/3

Solution (sequence of actions):

• Remember the rule for raising a number to a negative power. We get: 8 -1/3 = 1/(8) 1/3 .
• Note that the denominator is 8 to a fractional power. The general form of calculating a fractional degree is as follows: a m/n = n √8 m .
• Thus, 1/(8) 1/3 = 1/(3 √8 1). We get the cube root of eight, which is 2. Based on this, 1/(8) 1/3 = 1/(1/2) = 2.
• Answer: 8 -1/3 = 2

From school, we all know the rule about raising to a power: any number with an exponent N is equal to the result of multiplying this number by itself N times. In other words, 7 to the power of 3 is 7 multiplied by itself three times, that is, 343. Another rule - raising any value to the power of 0 gives one, and raising a negative value is the result of ordinary exponentiation, if it is even, and the same result with a minus sign if it is odd.

The rules also give an answer on how to raise a number to a negative power. To do this, you need to raise the required value by the module of the indicator in the usual way, and then divide the unit by the result.

From these rules, it becomes clear that the implementation of real tasks with large quantities will require the availability of technical means. Manually it will turn out to multiply by itself a maximum range of numbers up to twenty or thirty, and then no more than three or four times. This is not to mention the fact that then also divide the unit by the result. Therefore, for those who do not have a special engineering calculator at hand, we will tell you how to raise a number to a negative power in Excel.

Solving problems in Excel

To solve problems with exponentiation, Excel allows you to use one of two options.

The first is the use of the formula with the standard cap symbol. Enter the following data in the worksheet cells:

In the same way, you can raise the desired value to any power - negative, fractional. Let's do the following and answer the question of how to raise a number to a negative power. Example:

It is possible to correct directly in the formula =B2^-C2.

The second option is to use the ready-made "Degree" function, which takes two mandatory arguments - a number and an indicator. To start using it, it is enough to put an equal sign (=) in any free cell, indicating the beginning of the formula, and enter the above words. It remains to select two cells that will participate in the operation (or specify specific numbers manually), and press the Enter key. Let's look at a few simple examples.

			Formula	Result
			POWER(B2;C2)
			POWER(B3;C3)	0,002915

As you can see, there is nothing complicated about how to raise a number to a negative power and to a regular one using Excel. Indeed, to solve this problem, you can use both the familiar “lid” symbol and the built-in function of the program, which is easy to remember. This is a definite plus!

Let's move on to more complex examples. Let's recall the rule on how to raise a number to a negative power of a fractional character, and we will see that this task is very simply solved in Excel.

Fractional indicators

In short, the algorithm for calculating a number with a fractional exponent is as follows.

1. Convert a fractional exponent to a proper or improper fraction.
2. Raise our number to the numerator of the resulting converted fraction.
3. From the number obtained in the previous paragraph, calculate the root, with the condition that the root indicator will be the denominator of the fraction obtained in the first stage.

Agree that even when operating with small numbers and proper fractions, such calculations can take a lot of time. It's good that the spreadsheet processor Excel does not care what number and to what degree to raise. Try solving the following example in an Excel worksheet:

Using the above rules, you can check and make sure that the calculation is correct.

At the end of our article, we will give in the form of a table with formulas and results several examples of how to raise a number to a negative power, as well as several examples with fractional numbers and powers.

Example table

Check the Excel worksheet for the following examples. For everything to work correctly, you need to use a mixed reference when copying the formula. Fix the number of the column containing the number being raised, and the number of the row containing the indicator. Your formula should look something like this: "=$B4^C$3".

Number / Degree

Please note that positive numbers (even non-integer ones) are calculated without problems for any exponents. There are no problems with raising any numbers to integers. But raising a negative number to a fractional power will turn out to be a mistake for you, since it is impossible to follow the rule indicated at the beginning of our article about raising negative numbers, because parity is a characteristic of an exclusively INTEGER number.

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

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

a-n = (1 / a n)

The properties of negative values ​​of powers of numbers are similar to powers with a positive exponent. Represented Equation a m / a n = a m-n can be fair as

« Nowhere, as in mathematics, the clarity and accuracy of the conclusion does not allow a person to get away from the answer by talking around the question.».

A. D. Alexandrov

at n more m , as well as 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 indicator of the degree b - desired numerical value, a - the base of the degree as a natural numerical value. Then determine the module, that is, the absolute value of a negative number, which acts as an exponent. Calculate the degree of the given number relative to the absolute number as an indicator. The value of the degree is found by dividing one by the resulting number.

Rice. one

Consider the power of a number with a negative fractional exponent. Imagine that the number a is any positive number, the numbers n and m - integers. By definition a , which is raised to the power - equals one divided by the same number with a positive degree (Fig. 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 began 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 to arithmetic problems, regardless of their specific meaning and initial numerical data. In India, back in the 6th-11th centuries, negative values ​​of 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 directions of segments. It was Descartes who suggested that the number raised to a power be displayed as a two-story formula a n .