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

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

From these rules it becomes clear that performing real tasks involving large quantities will require the presence of technical means. Manually you can multiply by yourself a maximum range of numbers up to twenty to thirty, and then no more than three or four times. This is not to mention then dividing one 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 involving exponentiation, Excel allows you to use one of two options.

The first is the use of a formula with a standard “lid” sign. Enter the following data into the worksheet cells:

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

You can correct =B2^-C2 directly in the formula.

The second option is to use the ready-made “Degree” function, which takes two required arguments - a number and an exponent. To start using it, just put the equal sign (=) in any free cell, indicating the beginning of the formula, and enter the above words. All that remains is 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
			DEGREE(B2;C2)
			DEGREE(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 power using Excel. After all, to solve this problem, you can use both the familiar “lid” symbol and the program’s built-in function, which is easy to remember. This is a definite plus!

Let's move on to more complex examples. Let's remember the rule about how to raise a number to a negative fractional power, and we will see that this problem is very easily solved in Excel.

Fractional indicators

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

1. Convert a fraction to a correct one 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 exponent of the root will be the denominator of the fraction obtained at 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 Excel spreadsheet processor doesn’t care what number is raised to what power. Try solving the following example on an Excel worksheet:

Using the above rules, you can check and make sure that the calculation was done correctly.

At the end of our article, we will present 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 of operating with fractional numbers and powers.

Example table

Check out the following examples in your Excel worksheet. 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 have approximately next view: "=$B4^C$3".

Number/Degree

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

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

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

Steps

Solving simple problems with degrees

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

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

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

1. Multiply the result (16 in our example) by 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)
   • Continue multiplying the result of the first two numbers by the next number until you get your final answer. To do this, multiply the first two numbers, and then multiply the resulting result by the next number in the sequence. This method is valid for any degree. In our example you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .

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

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

3. On your calculator, look for the key labeled "exp" or " x n (\displaystyle x^(n))", or "^". Using this key you will raise a number to a power. It is almost impossible to calculate a degree with a large indicator manually (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click “View” -> “Engineering”. To switch to normal mode, click “View” -> “Normal”.

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

   Addition, subtraction, multiplication of powers

   1. You can add and subtract degrees only if they have the same bases. If you need to add powers with the same bases and exponents, then you can replace the addition operation with the multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented in the form 1 ∗ 4 5 (\displaystyle 1*4^(5)); Thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply that degree and this number. In our example, raise 4 to the fifth power, and then multiply the resulting result by 2. Remember that the addition operation can be replaced by the multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:

      • 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
      • 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
      • 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
      • 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))

   2. When multiplying powers with the same basis their indicators are added up (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:

      When raising a power to a power, the exponents are multiplied. For example, a degree is given. Since exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The point of this rule is that you multiply by powers (x 2) (\displaystyle (x^(2))) on itself five times. Like this:

      • (x 2) 5 (\displaystyle (x^(2))^(5))
      • (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
      • Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))

   3. A power with a negative exponent should be converted to a fraction (reverse power). It doesn't matter if you don't know what a reciprocal degree is. If you are given a degree with a negative exponent, e.g. 3 − 2 (\displaystyle 3^(-2)), write this degree in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:

      When dividing degrees with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .

      • The power in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.

   4. Below are some expressions that will help you learn to solve problems with exponents. The expressions given cover the material presented in this section. To see the answer, simply select the empty space after the equals sign.

   Solving problems with fractional exponents

   A power with a fractional exponent (for example, ) is converted to a root operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x (\displaystyle (\sqrt (x))). Here 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”, that is x 4 (\displaystyle (\sqrt[(4)](x))) .

   1. If the exponent is an improper fraction, then the exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule of multiplying powers. For example, a degree is given. Convert such a power into a root whose power is equal to the denominator of the fractional exponent, and then raise this root to a power equal to the numerator of the fractional exponent. To do this, remember that 5 3 (\displaystyle (\frac (5)(3))) = (1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5). In our example:

      • x 5 3 (\displaystyle x^(\frac (5)(3)))
      • x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x)))
      • x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3))) = (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
   2. Some calculators have a button to calculate exponents (you must first enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
   3. Remember that any number to the first power is equal to itself, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, e.g. 5 ∗ 1 = 5 (\displaystyle 5*1=5) And 5 / 1 = 5 (\displaystyle 5/1=5).
   4. Know that the power 0 0 does not exist (such a power has no solution). If you try to solve such a degree on a calculator or on a computer, you will receive an error. But remember that any number in zero degree equals 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 increases greatly. So if the answer seems wrong to you, it may actually be correct. You can check this by plotting any exponential function eg 2 x .

First level

Degree and its properties. Comprehensive guide (2019)

Why are degrees needed? Where will you need them? Why should you take the time to study them?

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

And, of course, knowledge of degrees will bring you closer to successful completion OGE or Unified State Exam and admission to the university of your dreams.

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

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

FIRST LEVEL

Exponentiation is a mathematical operation just like addition, subtraction, multiplication or division.

Now I will explain everything in human language using very simple examples. Be careful. The 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. Everyone has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written differently: . Mathematicians are cunning and lazy people. They first notice some patterns, and then figure out a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of cola bottles 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, more difficult and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful one:

What other clever counting tricks have lazy mathematicians come up with? Right - 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 that number to the fifth power. For example, . Mathematicians remember that two to the fifth power is... And they solve such problems in their heads - faster, easier and without mistakes.

All you need to do is remember what is highlighted in color in the table of powers of numbers. Believe me, this will make your life a lot easier.

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

Real life example #1

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

Imagine a square pool measuring one meter by one meter. The pool is at your dacha. It's hot and I really want to swim. But... the pool has no bottom! You need 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 bottom area of ​​the pool.

You can simply calculate by pointing your finger that the bottom of the pool consists of meter by meter cubes. If you have tiles one meter by one meter, you will need pieces. It's easy... But where have you seen such tiles? The tile will most likely be cm by cm. And then you will be tortured 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. Multiply by and you get tiles ().

Did you notice that to determine the area of ​​the pool bottom we multiplied the same number by itself? What does it mean? Since we are multiplying the same number, 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 them to a power is much easier and there are also fewer errors in calculations. For the Unified State Exam, this is very important).
So, thirty to the second power will be (). Or we 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 there are on the chessboard using the square of the number... On one side of the cells and on the other too. To calculate 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. You will 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 measuring a meter and a depth of a meter and try to count how many cubes measuring a meter by a meter will fit into your pool.

Just point your finger and count! One, two, three, four...twenty-two, twenty-three...How many did you get? Not 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 simplified this too. We reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself... What does this mean? This means you can take advantage of the degree. So, what you once counted with your finger, they do in one action: three cubed is equal. It is written like this: .

All that remains is remember 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 continue to count with your finger.

Well, to finally convince you that degrees were invented by quitters and cunning people to solve their own 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, for every million you make, you make another million. That is, every million you have doubles at the beginning of each year. How much money will you have in years? If you are sitting now and “counting with your finger,” it means you are very hardworking man 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 multiplied by two... in the second year - what happened, by two more, in the third year... Stop! You noticed that the number is multiplied by itself times. So two to the fifth power is a million! Now imagine that you have a competition and the one who can count the fastest will get these millions... It’s worth remembering the powers of numbers, don’t you think?

Real life example #5

You have a million. At the beginning of each year, for every million you make, you earn two more. Great isn't it? 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 to the fourth power it is equal to a million. You just have 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 a lot 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 an exponent? It's very simple - it's 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 degree basis? Even simpler - this is the number that is located below, at the base.

Here's a drawing for good measure.

Well in general view, in order to generalize and better remember... A degree with a base “ ” and an exponent “ ” is read as “to the degree” and is written as follows:

Power of a number with natural exponent

You probably already guessed: because the exponent is natural number. Yes, but what is it natural number? Elementary! Natural numbers are those numbers that are used in counting when listing objects: one, two, three... When we count objects, we do not say: “minus five,” “minus six,” “minus seven.” We also do not say: “one third”, or “zero point five”. These are not natural numbers. What numbers do you think these 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 number. Zero is easy to understand - it is when there is nothing. What do negative (“minus”) numbers mean? But they were invented primarily to indicate 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 arise, do you think? Very simple. Several thousand years ago, our ancestors discovered that they lacked 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, it's an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Summary:

Let us define the concept of a degree whose exponent is a natural number (i.e., integer and positive).

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

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

Properties of degrees

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

Let's see: what is it And ?

A-priory:

How many multipliers are there in total?

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

But by definition, this is a power of a number with an exponent, that is: , which is what needed to be proven.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule Necessarily there must be the same reasons!
Therefore, we combine the powers with the base, but it remains a separate factor:

only for the product of powers!

Under no circumstances can you write that.

2. that's it th power of a number

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

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

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total:

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

But this is not true, after all.

Power with negative base

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

But what should be the basis?

In powers of natural indicator the basis may be any number. Indeed, we can multiply any numbers by each other, be they positive, negative, or even.

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

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

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

Determine for yourself what sign the following expressions will have:

1)	2)	3)
4)	5)	6)

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: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive.

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

Example 6) is no longer so simple!

6 examples to practice

Analysis of the solution 6 examples

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

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

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

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses.

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 call 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, let us ask ourselves: why is this so?

Let's consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply 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 will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we cannot not only divide by zero, but also raise it to the zero power.

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

From here it’s easy to express what you’re looking for:

Now let’s extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number to a negative power is the reciprocal of the same number to positive degree. But at the same time The base cannot be null:(because you can’t divide by).

Let's summarize:

I. The expression is not defined in the case. If, then.

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

III. A number 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 independent solutions:

Analysis of problems for independent solution:

I know, I know, the numbers are scary, but on the Unified State Exam you have to be prepared for anything! Solve these examples or analyze their solutions if you couldn’t solve them and you will learn to cope with them easily in the exam!

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

Now let's consider rational numbers. What numbers are called rational?

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

To understand what it is "fractional degree", consider the fraction:

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

Now let's remember the rule about "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 to.

That is, the root of the th power is the inverse operation of raising to a power: .

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

Now we add the numerator: what is it? The answer is easy to obtain using the power-to-power rule:

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

None!

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

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 the expression?

But here a problem arises.

The number can be represented in the form of other, reducible fractions, for example, or.

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

Or another example: once, then you can write it down. But if we write down the indicator differently, we will again get into trouble: (that is, we got a completely different result!).

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

So if:

• - natural number;
• - integer;

Examples:

Rational exponents are very useful for transforming expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

Well, now comes the hardest part. Now we'll figure it out degree with irrational exponent.

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 natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms.

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

...number to the zeroth power- this is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number;

...negative integer degree- it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

By the way, in science a degree with a complex indicator is often used, that is, an indicator is not even 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 to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

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

Now look at the indicator. Doesn't he remind you of anything? Let us recall the formula for abbreviated multiplication of difference of squares:

In this case,

It turns out that:

Answer: .

2. We reduce fractions in exponents to same look: either both decimal or both regular. We get, for example:

Answer: 16

3. Nothing special, let’s use it normal properties degrees:

ADVANCED LEVEL

Determination of degree

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

• — degree base;
• - exponent.

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

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

Degree with an integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

Construction to the zero degree:

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 negative integer number:

(because you can’t divide by).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Power with rational exponent

• - natural number;
• - integer;

Examples:

Properties of degrees

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 we get the following product:

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

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule Necessarily there must be the same reasons. Therefore, we combine the powers with the base, but it remains a separate factor:

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

Under no circumstances can you write that.

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

Let's regroup this work like this:

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

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total: !

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

Power with a negative base.

Up to this point we have only discussed what it should be like index degrees. But what should be the basis? In powers of natural indicator the basis may be any number .

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

For example, is the number positive or negative? A? ?

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

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for 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. We can formulate the following simple rules:

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

Determine for yourself what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

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: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive. Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If we remember that, it becomes clear that, and therefore the basis 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 by each other, divide them into pairs and get:

Before we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

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

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

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses. But it's important to remember: All signs change at the same time! You can’t replace it with by changing only one disadvantage we don’t like!

Let's go back to the example:

And again the formula:

So now the last rule:

How will we prove it? Of course, as usual: let’s expand on the concept of degree and simplify it:

Well, now let's open the brackets. How many letters are there in total? times by multipliers - what does this remind you of? This is nothing more than a definition of an operation multiplication: There were only multipliers there. That is, this, by definition, is a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about degrees for the average level, we will analyze the degree with an irrational exponent. 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 numbers).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms. For example, a degree with a natural exponent is a number multiplied by itself several times; a number to the zero power is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number; a degree with an integer negative exponent - it’s as if some “reverse process” had 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). It is rather a purely mathematical object that mathematicians created to extend the concept of degree to the entire space of numbers.

By the way, in science a degree with a complex exponent is often used, that is, the 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:

1)

2)

3)

Answers:

1. Let's remember the difference of squares formula. Answer: .
2. We reduce the fractions to the same form: either both decimals or both ordinary ones. We get, for example: .
3. Nothing special, we use the usual properties of degrees:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree called an expression of the form: , where:

Degree with an integer exponent

a degree whose exponent is a natural number (i.e., integer and positive).

Power with rational exponent

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

Degree with irrational exponent

a degree whose exponent is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

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

NOW YOU HAVE THE WORD...

How do you like the article? Write below in the comments whether you liked it or not.

Tell us about your experience using degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck on your exams!

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

Yandex.RTB R-A-339285-1

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

Definition 1

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

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

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

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

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

Example 1

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

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

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

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

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

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

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

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

Definition 2

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

In this case, n is any positive integer.

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

Definition 3

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

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

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

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

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

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

Example 2

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

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

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

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

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

Example 3

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

Let us illustrate our idea with specific examples:

Example 4

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

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

Definition 4

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

What are powers with a rational exponent?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Definition 5

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

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

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

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

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

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

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

Definition 6

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

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

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

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

What are powers with irrational and real exponents?

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

Example 5

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

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

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

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

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

Definition 7

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

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

Number c is n-th power of a number a When:

Operations with degrees.

1. By multiplying degrees with the same base, their indicators are added:

a m·a n = a m + n .

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

3. Power of the product of 2 or more factors is equal to the product of the powers 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:

(a m) n = a m n .

Each formula above is true 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 a 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 radical number to this power:

4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:

5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:

A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power 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 with m< n.

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

To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.

A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.

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

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