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 all 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 a successful passing the OGE or the Unified State Examination and to enter 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'll explain everything human language very simple examples. Be careful. 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? 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 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 the 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. Unexpectedly, right?) Draw a pool: a bottom one meter in size and a meter deep and try to calculate how many cubes meter by meter in total 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 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 and in general view to generalize and remember better ... A degree with a base "" and an exponent "" is read as "to the degree" and is written as follows:

Power of a number with a natural exponent

You probably already guessed: because the exponent is 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, endless decimal. 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).

Any number to the first power is equal to itself:
To square a number is to multiply it by itself:
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.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule Necessarily must be same grounds!
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? A? ? 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:

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: 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 even degree 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 fractional degree 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 “number blank”, namely the 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.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule Necessarily must be on 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 index 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? A? ?

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. It is possible to formulate such simple rules:

even degree, - number positive.
Negative number raised to odd degree, - number negative.
A positive number to any power is a positive number.
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: 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 :

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:

Remember the difference of squares formula. Answer: .
We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
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 one of the previous articles, we already mentioned the degree of a number. Today we will try to navigate in the process of finding its meaning. Scientifically speaking, we will figure out how to properly exponentiate. We will understand how this process is carried out, at the same time touching on all possible exponents: natural, irrational, rational, whole.

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

Concept definition.
Raising to negative art.
Whole score.
Raising a number to an irrational power.

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

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

After that, you can proceed directly to the rules of construction.

Raising to a negative power

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

110 \u003d 0.1 \u003d 1 * 10 in minus 1 st.,

1100 \u003d 0.01 \u003d 1 * 10 in minus 2 steps.,

11000 \u003d 0.0001 \u003d 1 * 10 minus 3 st.,

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

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

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

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

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

How to raise a number to a natural power

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

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

Raise -2 to the 4th tbsp.

We are dealing with a natural indicator. Accordingly, the course of the decision will be as follows: (-2) in Art. 4 = (-2)*(-2)*(-2)*(-2). Now it remains only to carry out the multiplication of integers: (-2) * (-2) * (-2) * (-2). We get 16.

Answer to the task:

(-2) in Art. 4=16.

Example:

Calculate the value: three point two sevenths squared.

This example is equal to the following product: three point two seventh times three point two seventh. Remembering how the multiplication of mixed numbers is carried out, we complete the construction:

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

Answer: 10 39/49

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

We start by rounding P to hundredths and get:

P squared \u003d (3.14) 2 \u003d 9.8596. However, if we reduce P to ten-thousandths, we get P = 3.14159. Then squaring gets a completely different number: 9.8695877281.

It should be noted here that in many problems there is no need to raise irrational numbers to a power. As a rule, the answer is entered either in the form of, in fact, a degree, for example, the root of 6 to the power of 3, or, if the expression allows, its transformation is carried out: the root of 5 to 7 degrees \u003d 125 root of 5.

How to raise a number to an integer power

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

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

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

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

Accordingly, the construction of any real number to zero art. will give one.

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

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

Example:

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

Solution process:

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

The denominator is calculated simply: two cubed;

3 = 2*2*2=8.

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

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

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

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

Solving problems in Excel

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

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

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

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

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

Formula

Result

POWER(B2;C2)

POWER(B3;C3)

0,002915

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

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

Fractional indicators

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

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

Agree that even when operating with small numbers and proper fractions, such calculations can take a lot of time. It's good that the spreadsheet Excel processor no matter what number and to what degree to raise. Try solving the following example in an Excel worksheet:

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

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

Example table

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

Number / Degree

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

In continuation of the conversation about the degree of a number, it is logical to deal with finding the value of the degree. This process has been named exponentiation. In this article, we will just study how exponentiation is performed, while touching on all possible exponents - natural, integer, rational and irrational. And by tradition, we will consider in detail the solutions to examples of raising numbers to various degrees.

Page navigation.

What does "exponentiation" mean?

Let's start by explaining what is called exponentiation. Here is the relevant definition.

Definition.

Exponentiation is to find the value of the power of a number.

Thus, finding the value of the power of a with the exponent r and raising the number a to the power of r is the same thing. For example, if the task is “calculate the value of the power (0.5) 5”, then it can be reformulated as follows: “Raise the number 0.5 to the power of 5”.

Now you can go directly to the rules by which exponentiation is performed.

Raising a number to a natural power

In practice, equality based on is usually applied in the form . That is, when raising the number a to a fractional power m / n, the root of the nth degree from the number a is first extracted, after which the result is raised to an integer power m.

Consider solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We show two solutions.

First way. By definition of degree with a fractional exponent. We calculate the value of the degree under the sign of the root, after which we extract the cube root: .

The second way. By definition of a degree with a fractional exponent and on the basis of the properties of the roots, the equalities are true . Now extract the root Finally, we raise to an integer power .

Obviously, the obtained results of raising to a fractional power coincide.

Answer:

Note that a fractional exponent can be written as a decimal fraction or a mixed number, in these cases it should be replaced by the corresponding ordinary fraction, and then exponentiation should be performed.

Example.

Calculate (44.89) 2.5 .

Solution.

We write the exponent in the form common fraction(if necessary, see the article): . Now we perform raising to a fractional power:

Answer:

(44,89) 2,5 =13 501,25107 .

It should also be said that raising numbers to rational powers is a rather laborious process (especially when the numerator and denominator of the fractional exponent contain enough big numbers), which is usually done using computer science.

In conclusion of this paragraph, we will dwell on the construction of the number zero to a fractional power. We gave the following meaning to the fractional degree of zero of the form: for we have , while zero to the power m/n is not defined. So, zero to a positive fractional power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions and 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the degree of a number with an irrational exponent. At the same time, in practical purposes it is usually enough to get the value of the degree up to some sign. We note right away that in practice this value is calculated using electronic computing technology, since manual raising to an irrational power requires a large number of cumbersome calculations. But nevertheless we will describe in general terms the essence of the actions.

To get an approximate value of the power of a with an irrational exponent, some decimal approximation of the exponent is taken, and the value of the exponent is calculated. This value is the approximate value of the degree of the number a with an irrational exponent. The more accurate the decimal approximation of the number is taken initially, the more accurate the degree value will be in the end.

As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of an irrational indicator: . Now let's raise 2 to a rational power of 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈ 2.250116. Thus, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of an irrational exponent, for example, , then we get a more accurate value of the original degree: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

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. Be careful. 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.

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? Right - raising a number to a power.

Raising a number to a power

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.

Real life example #2

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 one meter in size and a meter deep and try to calculate how many cubes measuring a meter by a meter will enter your pool.

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.

Real life example #4

Real life example #5

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

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?

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).

Any number to the first power is equal to itself:
To square a number is to multiply it by itself:
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.

Solution:

Example: Simplify the expression.

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

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: 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

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).

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:

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;