How to solve expressions with negative exponents. Degree - properties, rules, actions and formulas

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 one 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< 0,5. В случае с отрицательной степенью все обстоит наоборот. При возведении десятичного (дробного) числа в отрицательную степень, значение увеличивается.

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

Lesson and presentation on the topic: "Degree with a negative indicator. Definition and examples of problem solving"

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Determining the degree with a negative exponent

We know well that any number to the zero power is equal to one. $a^0=1$, $a≠0$.
The question arises, what happens if you raise a number to a negative power? For example, what would the number $2^(-2)$ be equal to?
The first mathematicians who asked this question decided that it was not worth reinventing the wheel, and it was good that all the properties of the degrees remain the same. That is, when multiplying powers with the same base, the exponents add up.
Let's consider this case: $2^3*2^(-3)=2^(3-3)=2^0=1$.
We got that the product of such numbers should give unity. The unit in the product is obtained by multiplying the reciprocals, that is, $2^(-3)=\frac(1)(2^3)$.

Such reasoning led to the following definition.
Definition. If $n$ is a natural number and $а≠0$, then the following equality holds: $a^(-n)=\frac(1)(a^n)$.

An important identity that is often used: $(\frac(a)(b))^(-n)=(\frac(b)(a))^n$.
In particular, $(\frac(1)(a))^(-n)=a^n$.

Solution examples

Decision.
Let's consider each term separately.
1. $2^(-3)=\frac(1)(2^3)=\frac(1)(2*2*2)=\frac(1)(8)$.
2. $(\frac(2)(5))^(-2)=(\frac(5)(2))^2=\frac(5^2)(2^2)=\frac(25) (4)$.
3. $8^(-1)=\frac(1)(8)$.
It remains to perform addition and subtraction operations: $\frac(1)(8)+\frac(25)(4)-\frac(1)(8)=\frac(25)(4)=6\frac(1) (4)$.
Answer: $6\frac(1)(4)$.

Example 2
Express the given number as a power of a prime number $\frac(1)(729)$.

Decision.
Obviously $\frac(1)(729)=729^(-1)$.
But 729 is not a prime number ending in 9. We can assume that this number is a power of three. Let's sequentially divide 729 by 3.
1) $\frac(729)(3)=243$;
2) $\frac(243)(3)=81$;
3) $\frac(81)(3)=27$;
4) $\frac(27)(3)=9$;
5) $\frac(9)(3)=3$;
6) $\frac(3)(3)=1$.
Six operations have been completed, which means: $729=3^6$.
For our task:
$729^{-1}=(3^6)^{-1}=3^{-6}$.
Answer: $3^(-6)$.

Example 3. Express the expression as a power: $\frac(a^6*(a^(-5))^2)((a^(-3)*a^8)^(-1))$.
Decision. The first operation is always done inside the brackets, then the multiplication $\frac(a^6*(a^(-5))^2)((a^(-3)*a^8)^(-1))=\frac (a^6*a^(-10))((a^5)^(-1))=\frac(a^((-4)))(a^((-5)))=a^ (-4-(-5))=a^(-4+5)=a$.
Answer: $a$.

Example 4. Prove the identity:
$(\frac(y^2 (xy^(-1)-1)^2)(x(1+x^(-1)y)^2)*\frac(y^2(x^(-2 )+y^(-2)))(x(xy^(-1)+x^(-1)y))):\frac(1-x^(-1) y)(xy^(-1 )+1)=\frac(x-y)(x+y)$.

Decision.
On the left side, consider each factor in parentheses separately.
1. $\frac(y^2(xy^(-1)-1)^2)(x(1+x^(-1)y)^2)=\frac(y^2(\frac(x )(y)-1)^2)(x(1+\frac(y)(x))^2) =\frac(y^2(\frac(x^2)(y^2)-2\ frac(x)(y)+1))(x(1+2\frac(y)(x)+\frac(y^2)(x^2)))=\frac(x^2-2xy+ y^2)(x+2y+\frac(y^2)(x))=\frac(x^2-2xy+y^2)(\frac(x^2+2xy+y^2)(x) )=\frac(x(x^2-2xy+y^2))((x^2+2xy+y^2))$.
2. $\frac(y^2(x^(-2)+y^(-2)))(x(xy^(-1)+x^(-1)y))=\frac(y^ 2(\frac(1)(x^2)+\frac(1)(y^2)))(x(\frac(x)(y)+\frac(y)(x))) =\frac (\frac(y^2)(x^2)+1)(\frac(x^2)(y)+y)=\frac(\frac(y^2+x^2)(x^2) )((\frac(x^2+y^2)(y)))=\frac(y^2+x^2)(x^2) *\frac(y)(x^2+y^2 )=\frac(y)(x^2)$.
3. $\frac(x(x^2-2xy+y^2))((x^2+2xy+y^2))*\frac(y)(x^2)=\frac(y(x ^2-2xy+y^2))(x(x^2+2xy+y^2))=\frac(y(x-y)^2)(x(x+y)^2)$.
4. Let's move on to the fraction by which we divide.
$\frac(1-x^(-1)y)(xy^(-1)+1)=\frac(1-\frac(y)(x))(\frac(x)(y)+1 )=\frac(\frac(x-y)(x))(\frac(x+y)(y))=\frac(x-y)(x)*\frac(y)(x+y)=\frac( y(x-y))(x(x+y))$.
5. Let's do the division.
$\frac(y(x-y)^2)(x(x+y)^2):\frac(y(x-y))(x(x+y))=\frac(y(x-y)^2)( x(x+y)^2)*\frac(x(x+y))(y(x-y))=\frac(x-y)(x+y)$.
We obtained the correct identity, which was required to be proved.

At the end of the lesson, we will write down the rules for actions with degrees again, here the exponent is an integer.
$a^s*a^t=a^(s+t)$.
$\frac(a^s)(a^t)=a^(s-t)$.
$(a^s)^t=a^(st)$.
$(ab)^s=a^s*b^s$.
$(\frac(a)(b))^s=\frac(a^s)(b^s)$.

Tasks for independent solution

First level

Degree and its properties. Comprehensive Guide (2019)

Why are degrees needed? Where do you need them? Why do you need to spend time studying them?

To learn everything about degrees, what they are for, how to use your knowledge in everyday life, read this article.

And, of course, knowing the degrees will bring you closer to successfully passing the OGE or the Unified State Examination and entering the university of your dreams.

Let's go... (Let's go!)

Important note! If instead of formulas you see gibberish, clear your cache. To do this, press CTRL+F5 (on Windows) or Cmd+R (on Mac).

FIRST LEVEL

Exponentiation is the same mathematical operation as addition, subtraction, multiplication or division.

Now I will explain everything in human language using very simple examples. Pay attention. Examples are elementary, but explain important things.

Let's start with addition.

There is nothing to explain here. You already know everything: there are eight of us. Each has two bottles of cola. How much cola? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written in a different way: . Mathematicians are cunning and lazy people. They first notice some patterns, and then come up with a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of bottles of cola and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table. Of course, you can do everything slower, harder and with mistakes! But…

Here is the multiplication table. Repeat.

And another, prettier one:

And what other tricky counting tricks did lazy mathematicians come up with? Correctly - raising a number to a power.

Raising a number to a power

If you need to multiply a number by itself five times, then mathematicians say that you need to raise this number to the fifth power. For example, . Mathematicians remember that two to the fifth power is. And they solve such problems in their mind - faster, easier and without errors.

To do this, you only need remember what is highlighted in color in the table of powers of numbers. Believe me, it will make your life much easier.

By the way, why is the second degree called square numbers, and the third cube? What does it mean? A very good question. Now you will have both squares and cubes.

Real life example #1

Let's start with a square or the second power of a number.

Imagine a square pool measuring meters by meters. The pool is in your backyard. It's hot and I really want to swim. But ... a pool without a bottom! It is necessary to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the area of ​​the bottom of the pool.

You can simply count by poking your finger that the bottom of the pool consists of cubes meter by meter. If your tiles are meter by meter, you will need pieces. It's easy... But where did you see such a tile? The tile will rather be cm by cm. And then you will be tormented by “counting with your finger”. Then you have to multiply. So, on one side of the bottom of the pool, we will fit tiles (pieces) and on the other, too, tiles. Multiplying by, you get tiles ().

Did you notice that we multiplied the same number by itself to determine the area of ​​the bottom of the pool? What does it mean? Since the same number is multiplied, we can use the exponentiation technique. (Of course, when you have only two numbers, you still need to multiply them or raise them to a power. But if you have a lot of them, then raising to a power is much easier and there are also fewer errors in calculations. For the exam, this is very important).
So, thirty to the second degree will be (). Or you can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. And vice versa, if you see a square, it is ALWAYS the second power of some number. A square is an image of the second power of a number.

Real life example #2

Here's a task for you, count how many squares are on the chessboard using the square of the number ... On one side of the cells and on the other too. To count their number, you need to multiply eight by eight, or ... if you notice that a chessboard is a square with a side, then you can square eight. Get cells. () So?

Real life example #3

Now the cube or the third power of a number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Unexpected, right?) Draw a pool: a bottom a meter in size and a meter deep and try to calculate how many meter by meter cubes will enter your pool.

Just point your finger and count! One, two, three, four…twenty-two, twenty-three… How much did it turn out? Didn't get lost? Is it difficult to count with your finger? So that! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes ... Easier, right?

Now imagine how lazy and cunning mathematicians are if they make that too easy. Reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself ... And what does this mean? This means that you can use the degree. So, what you once counted with a finger, they do in one action: three in a cube is equal. It is written like this:

Remains only memorize the table of degrees. Unless, of course, you are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can keep counting with your finger.

Well, in order to finally convince you that the degrees were invented by loafers and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Real life example #4

You have a million rubles. At the beginning of each year, you earn another million for every million. That is, each of your million at the beginning of each year doubles. How much money will you have in years? If you are now sitting and “counting with your finger”, then you are a very hardworking person and .. stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two times two ... in the second year - what happened, by two more, in the third year ... Stop! You noticed that the number is multiplied by itself once. So two to the fifth power is a million! Now imagine that you have a competition and the one who calculates faster will get these millions ... Is it worth remembering the degrees of numbers, what do you think?

Real life example #5

You have a million. At the beginning of each year, you earn two more for every million. It's great right? Every million is tripled. How much money will you have in a year? Let's count. The first year - multiply by, then the result by another ... It's already boring, because you already understood everything: three is multiplied by itself times. So the fourth power is a million. You just need to remember that three to the fourth power is or.

Now you know that by raising a number to a power, you will make your life much easier. Let's take a further look at what you can do with degrees and what you need to know about them.

Terms and concepts ... so as not to get confused

So, first, let's define the concepts. What do you think, what is exponent? It's very simple - this is the number that is "at the top" of the power of the number. Not scientific, but clear and easy to remember ...

Well, at the same time, what such a base of degree? Even simpler is the number that is at the bottom, at the base.

Here's a picture for you to be sure.

Well, in general terms, in order to generalize and remember better ... A degree with a base "" and an indicator "" is read as "in the degree" and is written as follows:

Power of a number with a natural exponent

You probably already guessed: because the exponent is a natural number. Yes, but what is natural number? Elementary! Natural numbers are those that are used in counting when listing items: one, two, three ... When we count items, we don’t say: “minus five”, “minus six”, “minus seven”. We don't say "one third" or "zero point five tenths" either. These are not natural numbers. What do you think these numbers are?

Numbers like "minus five", "minus six", "minus seven" refer to whole numbers. In general, integers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and a number. Zero is easy to understand - this is when there is nothing. And what do negative ("minus") numbers mean? But they were invented primarily to denote debts: if you have a balance on your phone in rubles, this means that you owe the operator rubles.

All fractions are rational numbers. How did they come about, do you think? Very simple. Several thousand years ago, our ancestors discovered that they did not have enough natural numbers to measure length, weight, area, etc. And they came up with rational numbers… Interesting, isn't it?

There are also irrational numbers. What are these numbers? In short, an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, then you get an irrational number.

Summary:

Let's define the concept of degree, the exponent of which is a natural number (that is, integer and positive).

1. Any number to the first power is equal to itself:
2. To square a number is to multiply it by itself:
3. To cube a number is to multiply it by itself three times:

Definition. To raise a number to a natural power is to multiply the number by itself times:
.

Degree properties

Where did these properties come from? I will show you now.

Let's see what is and ?

A-priory:

How many multipliers are there in total?

It's very simple: we added factors to the factors, and the result is factors.

But by definition, this is the degree of a number with an exponent, that is: , which was required to be proved.

Example: Simplify the expression.

Decision:

Example: Simplify the expression.

Decision: It is important to note that in our rule necessarily must be the same reason!
Therefore, we combine the degrees with the base, but remain a separate factor:

only for products of powers!

Under no circumstances should you write that.

2. that is -th power of a number

Just as with the previous property, let's turn to the definition of the degree:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In fact, this can be called "bracketing the indicator". But you can never do this in total:

Let's recall the formulas for abbreviated multiplication: how many times did we want to write?

But that's not true, really.

Degree with a negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the basis?

In degrees from natural indicator the basis may be any number. Indeed, we can multiply any number by each other, whether they are positive, negative, or even.

Let's think about what signs ("" or "") will have degrees of positive and negative numbers?

For example, will the number be positive or negative? BUT? ? With the first, everything is clear: no matter how many positive numbers we multiply with each other, the result will be positive.

But the negative ones are a little more interesting. After all, we remember a simple rule from the 6th grade: “a minus times a minus gives a plus.” That is, or. But if we multiply by, it turns out.

Determine for yourself what sign the following expressions will have:

Did you manage?

Here are the answers: In the first four examples, I hope everything is clear? We simply look at the base and exponent, and apply the appropriate rule.

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In example 5), everything is also not as scary as it seems: it doesn’t matter what the base is equal to - the degree is even, which means that the result will always be positive.

Well, except when the base is zero. The base is not the same, is it? Obviously not, since (because).

Example 6) is no longer so simple!

6 practice examples

Analysis of the solution 6 examples

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were swapped, the rule could apply.

But how to do that? It turns out that it is very easy: the even degree of the denominator helps us here.

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

whole we name the natural numbers, their opposites (that is, taken with the sign "") and the number.

positive integer, and it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at new cases. Let's start with an indicator equal to.

Any number to the zero power is equal to one:

As always, we ask ourselves: why is this so?

Consider some power with a base. Take, for example, and multiply by:

So, we multiplied the number by, and got the same as it was -. What number must be multiplied by so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you still get zero, this is clear. But on the other hand, like any number to the zero degree, it must be equal. So what is the truth of this? Mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we can not only divide by zero, but also raise it to the zero power.

Let's go further. In addition to natural numbers and numbers, integers include negative numbers. To understand what a negative degree is, let's do the same as last time: we multiply some normal number by the same in a negative degree:

From here it is already easy to express the desired:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate the rule:

A number to a negative power is the inverse of the same number to a positive power. But at the same time base cannot be null:(because it is impossible to divide).

Let's summarize:

I. Expression is not defined in case. If, then.

II. Any number to the zero power is equal to one: .

III. A number that is not equal to zero to a negative power is the inverse of the same number to a positive power: .

Tasks for independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are scary, but at the exam you have to be ready for anything! Solve these examples or analyze their solution if you couldn't solve it and you will learn how to easily deal with them in the exam!

Let's continue to expand the range of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is "fractional degree" Let's consider a fraction:

Let's raise both sides of the equation to a power:

Now remember the rule "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal.

That is, the root of the th degree is the inverse operation of exponentiation: .

It turns out that. Obviously, this special case can be extended: .

Now add the numerator: what is it? The answer is easy to get with the power-to-power rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But here a problem arises.

The number can be represented as other, reduced fractions, for example, or.

And it turns out that it exists, but does not exist, and these are just two different records of the same number.

Or another example: once, then you can write it down. But as soon as we write the indicator in a different way, we again get trouble: (that is, we got a completely different result!).

To avoid such paradoxes, consider only positive base exponent with fractional exponent.

So if:

• - natural number;
• is an integer;

Examples:

Powers with a rational exponent are very useful for transforming expressions with roots, for example:

5 practice examples

Analysis of 5 examples for training

Well, now - the most difficult. Now we will analyze degree with an irrational exponent.

All the rules and properties of degrees here are exactly the same as for degrees with a rational exponent, with the exception of

Indeed, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms.

For example, a natural exponent is a number multiplied by itself several times;

...zero power- this is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number;

...negative integer exponent- it’s as if a certain “reverse process” has taken place, that is, the number was not multiplied by itself, but divided.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number.

But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a degree to a degree:

Now look at the score. Does he remind you of anything? We recall the formula for abbreviated multiplication of the difference of squares:

In this case,

It turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal or both ordinary. We get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of degrees:

ADVANCED LEVEL

Definition of degree

The degree is an expression of the form: , where:

• — base of degree;
• - exponent.

Degree with natural exponent (n = 1, 2, 3,...)

Raising a number to the natural power n means multiplying the number by itself times:

Power with integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

erection to zero power:

The expression is indefinite, because, on the one hand, to any degree is this, and on the other hand, any number to the th degree is this.

If the exponent is integer negative number:

(because it is impossible to divide).

One more time about nulls: the expression is not defined in the case. If, then.

Examples:

Degree with rational exponent

• - natural number;
• is an integer;

Examples:

Degree properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression, the following product is obtained:

But by definition, this is a power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

Decision : It is important to note that in our rule necessarily must have the same basis. Therefore, we combine the degrees with the base, but remain a separate factor:

Another important note: this rule - only for products of powers!

Under no circumstances should I write that.

Just as with the previous property, let's turn to the definition of the degree:

Let's rearrange it like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the -th power of the number:

In fact, this can be called "bracketing the indicator". But you can never do this in total:!

Let's recall the formulas for abbreviated multiplication: how many times did we want to write? But that's not true, really.

Power with a negative base.

Up to this point, we have discussed only what should be indicator degree. But what should be the basis? In degrees from natural indicator the basis may be any number .

Indeed, we can multiply any number by each other, whether they are positive, negative, or even. Let's think about what signs ("" or "") will have degrees of positive and negative numbers?

For example, will the number be positive or negative? BUT? ?

With the first, everything is clear: no matter how many positive numbers we multiply with each other, the result will be positive.

But the negative ones are a little more interesting. After all, we remember a simple rule from the 6th grade: “a minus times a minus gives a plus.” That is, or. But if we multiply by (), we get -.

And so on ad infinitum: with each subsequent multiplication, the sign will change. You can formulate these simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any power is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself what sign the following expressions will have:

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We simply look at the base and exponent, and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it doesn’t matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, except when the base is zero. The base is not the same, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and divide them into each other, divide them into pairs and get:

Before analyzing the last rule, let's solve a few examples.

Calculate the values ​​of expressions:

Solutions :

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares!

We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were reversed, rule 3 could be applied. But how to do this? It turns out that it is very easy: the even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it looks like this:

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change at the same time! It cannot be replaced by by changing only one objectionable minus to us!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Well, now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing but the definition of an operation multiplication: total there turned out to be multipliers. That is, it is, by definition, a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about the degrees for the average level, we will analyze the degree with an irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number; a degree with an integer negative indicator - it is as if a certain “reverse process” has occurred, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians have created to extend the concept of a degree to the entire space of numbers.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number. But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

Answers:

1. Remember the difference of squares formula. Answer: .
2. We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
3. Nothing special, we apply the usual properties of degrees:

SECTION SUMMARY AND BASIC FORMULA

Degree is called an expression of the form: , where:

Degree with integer exponent

degree, the exponent of which is a natural number (i.e. integer and positive).

Degree with rational exponent

degree, the indicator of which is negative and fractional numbers.

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree properties

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any power is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE A WORD...

How do you like the article? Let me know in the comments below if you liked it or not.

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

In the framework of this material, we will analyze what a power of a number is. In addition to the basic definitions, we will formulate what degrees with natural, integer, rational and irrational exponents are. As always, all concepts will be illustrated with examples of tasks.

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First, we 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 the time being we will take a real number as a base (let us denote it by the letter a), and as an indicator - a natural number (denoted by the letter n).

Definition 1

The power of a with a 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 the degree is a convenient form of writing a large number of equal factors. So, a record of the form 8 8 8 8 can be reduced 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 analyzed this in the article devoted to the multiplication of natural numbers.

How to correctly read the record of the degree? The generally accepted option is "a to the power of n". Or you can say "the nth power of a" or "the nth power". If, say, in the example there is an 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 degrees of the number have their own well-established names: square and cube. If we see the second power, for example, of 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 is the "cube of the number 5" or "5 cubed". However, it is also possible to use the standard wording “in the second / third degree”, this will not be a mistake.

Example 1

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

The base does not have to be an integer: for the degree (4 , 32) 9 the base will be a fraction 4, 32, and the exponent will be nine. Pay attention to the brackets: such a notation is made for all degrees, the bases of which differ from natural numbers.

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

What are the brackets for? They help to avoid errors in calculations. Let's say we have two entries: (− 2) 3 and − 2 3 . The first of them 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 degree of a number - a^n(where a is the base and n is the exponent). So 4^9 is the same as 4 9 . If n is a multi-digit number, it is enclosed in parentheses. For example, 15 ^ (21) , (− 3 , 1) ^ (156) . But we will use the notation a n as more common.

How to calculate the value of a degree with a natural exponent is easy to guess from its definition: you just need to multiply a n -th number of times. We wrote more about this in another article.

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

The exponents can contain 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 degree of a number with a positive integer exponent can be displayed as a formula: .

Moreover, n is any positive integer.

Let's deal with the concept of zero degree. To do this, we use an approach that takes into account the property of the quotient 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 dividing by zero. If the values ​​of m and n are equal, then we will get the following result: a n: a n = a n − n = a 0

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

However, such a proof is not suitable for zero to the power zero. 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 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 0 0 , that is, it can be equal to any number, and this will not affect the validity of the equality. Therefore, a record of the form 0 0 has no special meaning of its own, and we will not ascribe 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 equal to zero. Thus, the degree of any non-zero number with a zero exponent is equal to 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, it remains for us to figure out what a negative degree is. To do this, we need the same property of the product of powers with equal bases, which we have already used above: a m · a n = a m + n.

We introduce the condition: m = − n , then a must 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 a negative integer power is nothing but a fraction 1 a n .

This formulation confirms that for a degree with a negative integer 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

The power a with a negative integer n can be represented as a fraction 1 a n . Thus, a - n = 1 a n under the condition a ≠ 0 and n is any natural number.

Let's 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 with natural exponent z is: a z = a z , e c and z is a positive integer 1 , z = 0 and a ≠ 0 , (if z = 0 and a = 0 we get 0 0 , the values ​​of the expression 0 0 are not is determined)   1 a z , if z is a negative integer and a ≠ 0 ( if z is a negative integer and a = 0 we get 0 z , it is a n d e n t i o n )

What are degrees with a rational exponent

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

What are rational numbers? Their set includes both integer and fractional numbers, while fractional numbers can be represented as ordinary fractions (both positive and negative). We formulate the definition of the degree 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 property to hold in a degree, the equality a m n n = a m n · n = a m must be true.

Given the definition of an 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 the degree with an integer exponent will be true under the condition a m n = a m n .

The main conclusion from our reasoning is as follows: the degree of some number a with a fractional exponent m / n is the root of the nth degree from the number a to the power m. This is true if, for given values ​​of m, n, and a, the expression a m n makes sense.

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

The fractional exponent m/n for some positive number a is the nth root of a raised to the m power. In the form of a formula, this can be represented as follows:

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

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

0 m n = 0 m n = 0 under the condition of positive integer m and natural n .

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

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

The expression a m n sometimes still makes sense for some negative values ​​of a and some negative values ​​of m . So, the entries are correct (- 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 need to introduce one more condition: the degree a, in the exponent of which there is a reducible ordinary fraction, is considered 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 a record 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 m is positive and a is any non-negative number, then a m n makes sense. The condition for a non-negative a is necessary, since the root of an even degree is not extracted from a negative number. If the value of m is positive, then a can be both negative and zero, because An odd root can be taken from any real number.

Let's combine all the data above the definition 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 reduced fraction m · k n · k, the degree can be replaced by a m n .

The degree of 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 values ​​of 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 powers: - 2 11 6 , - 2 1 2 3 2 , 0 - 2 5 .

Now let's explain the importance of the condition mentioned above: why replace a fraction with a reducible exponent for a fraction with an irreducible one. If we would not have done this, then such situations would have turned out, say, 6 / 10 = 3 / 5. Then (- 1) 6 10 = - 1 3 5 should be true, 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 the degree with a fractional exponent, which we gave first, is more convenient to apply in practice than the second, so we will continue to use it.

Definition 6

Thus, the power of a positive number a with fractional exponent m / n is defined as 0 m n = 0 m n = 0 . In case of negative a the notation a m n makes no sense. Degree 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 the conclusions, we note that any fractional indicator can be written both as a mixed number and as a decimal fraction: 5 1, 7, 3 2 5 - 2 3 7.

When calculating, it is better to replace the exponent with an ordinary fraction and then use the definition of the 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 degrees with irrational and real exponent

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. About rational we have already mentioned above. Let's deal with irrational indicators step by step.

Example 5

Suppose 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 powers a a 0 , a a 1 , a a 2 , . . . . If we recall what we talked about earlier about raising numbers to a rational power, then we can calculate the values ​​​​of these powers ourselves.

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 degrees can be reduced to a number, which will be the value of the degree with the base a and the 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 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, while 0 a \u003d 0 So, 0 6 \u003d 0, 0 21 3 3 \u003d 0. And for negative ones, this cannot be done, 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 equal 1 .

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Power formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is an n-th power of a number a when:

Operations with degrees.

1. Multiplying degrees with the same base, their indicators add up:

a ma n = a m + n .

2. In the division of degrees with the same base, their indicators are subtracted:

3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n b n c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n / b n .

5. Raising a power to a power, the exponents are multiplied:

(am) n = a m n .

Each formula above is correct in the directions from left to right and vice versa.

for example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the root number to this power:

4. If we increase the degree of the root in n once and at the same time raise to n th power is a root number, then the value of the root will not change:

5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:

Degree with a negative exponent. The degree of a number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to the absolute value of the non-positive exponent:

Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.

for example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.

Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.

for example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

A degree with a fractional exponent. To raise a real number a to a degree m/n, you need to extract the root n th degree of m th power of this number a.