What is called the base of the degree. What is a power of a number

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

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Power 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 power of a number a with natural exponent n is given for a, which we will call degree basis, and n, which we will call exponent. We also note that a degree with a natural exponent is determined through a product, so to understand the material below you need to have an understanding of multiplying numbers.

Definition.

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

It’s worth mentioning right away about the rules for reading degrees. The universal way to read the notation a n is: “a to the power of n”. In some cases, the following options are also acceptable: “a to the nth power” and “nth power of a”. For example, let's take the power 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 square the number, for example, 7 2 is read as “seven squared” or “the square of the number seven.” The third power of a number is called cubed numbers, for example, 5 3 can be read as “five cubed” or you can say “cube of the number 5”.

It's time to bring examples of degrees with natural exponents. Let's start with the degree 5 7, here 5 is the base of the degree, 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 power 4.32 is written in parentheses: to avoid discrepancies, we will put in parentheses all bases of the power that are different from natural numbers. As an example, we give the following degrees with natural exponents , 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 records of the form (−2) 3 and −2 3. The expression (−2) 3 is a power of −2 with a natural exponent of 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .

Note that there is a notation for the power of a number a with an exponent n of the form a^n. Moreover, if n is a multi-valued 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 some more examples of writing degrees using the symbol “^”: 14^(21) , (−2,1)^(155) . In what follows, we will primarily use degree notation of the form a n .

One of the problems inverse to raising to a power with a natural exponent is the problem of finding the base of the power from a known value of the power and a known exponent. This task leads to .

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

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

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

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

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

The easiest way is to impose a constraint on a by taking a≥0 for positive m and a>0 for negative m (since for m≤0 the degree 0 of m is not defined). Then we get the following definition of a 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 nth root of the number a to the power m, that is, .

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

Definition.

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

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

Another approach to determining a degree with a fractional exponent m/n is to separately consider even and odd exponents of the root. This approach requires an additional condition: the power of the number a, the exponent of which is , is considered to be the power of the number a, the exponent of which is the corresponding irreducible fraction (we will explain the importance of this condition 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 (an even root of a negative number does not make sense); for negative m, the number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be non-zero (so that there is no division by zero).

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

Definition.

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

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

“Comparative degree” - A ferret lived in the same hole. N.f. Smart + MORE - smarter N.f. Smart + LESS - less smart. Role in a sentence. Our less nimble dogs go to cheer for the mice at the races. Municipal educational institution "Elgai basic secondary school". A hamster is more nimble than a puppy. Somehow our shoe was dragged away by a less nimble neighbor’s puppy.

“Degree with a natural indicator” - Degree with a natural and integer indicator. (-1)2k=1, (-1)2k-1= -1. Properties of degrees with natural exponents. Determination of degree with a natural indicator. 1 to any power is equal to 1 1n=1. What is a degree? How to write in short. Multiplication of powers with the same bases. N terms. 10n=100000…0.

“Degree with an integer exponent” - Calculate. Express the expression as a power. Express the expression x-12 as the product of two powers with base x if one factor is known. Arrange in descending order. Simplify. For what values of x is the equality true?

“Equations of the third degree” - (In the third case - the minimum, in the fourth - the maximum). In the first and second cases we say that the function is monotonic at the point x =. Our formula yields: “Great Art.” So, Tartaglia allowed himself to be persuaded. Lemma. In the third and fourth cases we say that the function has an extremum at the point x =. Opening the parentheses.

“Properties of a degree” - Generalization of knowledge and skills in applying the properties of a degree with a natural indicator. Properties of degrees with natural exponents. Brainstorming. The cube of what number is 64? Computational pause. Properties of degrees with natural exponents. Development of perseverance, mental activity and creative activity.

“Root of the nth degree” - Definition 2: A). Let's cube both sides of the equation: - Radical expression. Consider the equation x? = 1. Let’s raise both sides of the equation to the fourth power: Let’s plot the functions y = x? and y = 1. The concept of the nth root of a real number. If n is odd, then one root: Let's build graphs of the functions y = x? and y = 1.

Please note that this section discusses the concept degrees with natural exponent only and zero.

The concept and properties of powers with rational exponents (with negative and fractional) will be discussed in lessons for grade 8.

So, let's figure out what a power of a number is. To write the product of a number by itself, the abbreviated notation is used several times.

Instead of the product of six identical factors 4 · 4 · 4 · 4 · 4 · 4, write 4 6 and say “four to the sixth power”.

4 4 4 4 4 4 = 4 6

The expression 4 6 is called a power of a number, where:

4 — degree base;
6 — exponent.

In general, a degree with a base “a” and exponent “n” is written using the expression:

Remember!

The power of a number “a” with a natural exponent “n” greater than 1 is the product of “n” identical factors, each of which is equal to the number “a”.

The entry “a n” reads like this: “a to the power of n” or “nth power of the number a”.

The exceptions are the following entries:

a 2 - it can be pronounced as “a squared”;
a 3 - it can be pronounced as “a cubed.”

a 2 - “a to the second power”;
a 3 - “a to the third power.”

Special cases arise if the exponent is equal to one or zero (n = 1; n = 0).

Remember!

The power of the number “a” with exponent n = 1 is this number itself:
a 1 = a

Any number to the zero power is equal to one.
a 0 = 1

Zero to any natural power is equal to zero.
0 n = 0

One to any power is equal to 1.
1 n = 1

Expression 0 0 ( zero to the zero power) are considered meaningless.

(−32) 0 = 1
0 253 = 0
1 4 = 1

When solving examples, you need to remember that raising to a power is finding a numerical or letter value after raising it to a power.

Example. Raise to a power.

5 3 = 5 5 5 = 125
2.5 2 = 2.5 2.5 = 6.25
( · = = 81
256

Raising a negative number to the power

The base (the number that is raised to the power) can be any number—positive, negative, or zero.

Remember!

Raising a positive number to a power produces a positive number.

When zero is raised to a natural power, the result is zero.

When a negative number is raised to a power, the result can be either a positive number or a negative number. It depends on whether the exponent was an even or odd number.

Let's look at examples of raising negative numbers to powers.

From the examples considered, it is clear that if a negative number is raised to an odd power, then a negative number is obtained. Since the product of an odd number of negative factors is negative.

If a negative number is raised to an even power, it becomes a positive number. Since the product of an even number of negative factors is positive.

Remember!

A negative number raised to an even power is a positive number.

A negative number raised to an odd power is a negative number.

The square of any number is a positive number or zero, that is:

a 2 ≥ 0 for any a.

2 · (−3) 2 = 2 · (−3) · (−3) = 2 · 9 = 18
−5 · (−2) 3 = −5 · (−8) = 40

Pay attention!

When solving examples of exponentiation, mistakes are often made, forgetting that the entries (−5) 4 and −5 4 are different expressions. The results of raising these expressions to powers will be different.

Calculating (−5) 4 means finding the value of the fourth power of a negative number.

(−5) 4 = (−5) · (−5) · (−5) · (−5) = 625

While finding “−5 4” means that the example needs to be solved in 2 steps:

Raise the positive number 5 to the fourth power.
5 4 = 5 5 5 5 = 625
Place a minus sign in front of the result obtained (that is, perform a subtraction action).
−5 4 = −625

Example. Calculate: −6 2 − (−1) 4

−6 2 − (−1) 4 = −37

6 2 = 6 6 = 36
−6 2 = −36
(−1) 4 = (−1) · (−1) · (−1) · (−1) = 1
−(−1) 4 = −1
−36 − 1 = −37

Procedure in examples with degrees

Calculating a value is called the action of exponentiation. This is the third stage action.

Remember!

In expressions with powers that do not contain parentheses, first do exponentiation, then multiplication and division, and at the end addition and subtraction.

If the expression contains parentheses, then first perform the actions in the parentheses in the order indicated above, and then perform the remaining actions in the same order from left to right.

Example. Calculate:

To make it easier to solve examples, it is useful to know and use the table of degrees, which you can download for free on our website.

To check your results, you can use the calculator on our website "