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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

Raising to a power is the same mathematical operation like addition, subtraction, multiplication or division.

Now I'll explain everything human language 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,” 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 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 a 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 for measuring length, weight, area, etc. And they came up with rational numbers... Interesting, isn't it?

Is there some more 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? ? The first one is clear: no matter how much positive numbers We didn’t 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 in last time: multiply some normal number by the same number to a negative power:

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!

Remember the rule: any number raised to even degree- the number is positive. 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:

Degrees with rational indicator very useful for converting 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;

...degree with integer negative indicator - 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 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, we use the usual properties of 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 is - 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, which means 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 by each other, divide them into pairs and get:

Before you take it apart 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!

The calculator helps you quickly raise a number to a power online. The base of the degree can be any number (both integers and reals). The exponent can also be an integer or real, and can also be positive or negative. Keep in mind that for negative numbers, raising to a non-integer power is undefined, so the calculator will report an error if you attempt it.

Degree calculator

Raise to power

Exponentiations: 20880

What is a natural power of a number?

The number p is called the nth power of a number if p is equal to the number a multiplied by itself n times: p = a n = a·...·a
n - called exponent, and the number a is degree basis.

How to raise a number to a natural power?

To understand how to build different numbers to natural powers, consider a few examples:

Example 1. Raise the number three to the fourth power. That is, it is necessary to calculate 3 4
Solution: as mentioned above, 3 4 = 3·3·3·3 = 81.
Answer: 3 4 = 81 .

Example 2. Raise the number five to the fifth power. That is, it is necessary to calculate 5 5
Solution: similarly, 5 5 = 5·5·5·5·5 = 3125.
Answer: 5 5 = 3125 .

Thus, to raise a number to a natural power, you just need to multiply it by itself n times.

What is a negative power of a number?

The negative power -n of a is one divided by a to the power of n: a -n = .

In this case, a negative power exists only for non-zero numbers, since otherwise division by zero would occur.

How to raise a number to a negative integer power?

To raise a non-zero number to a negative power, you need to calculate the value of this number to the same positive power and divide one by the result.

Example 1. Raise the number two to the negative fourth power. That is, you need to calculate 2 -4

Solution: as stated above, 2 -4 = = = 0.0625.

Answer: 2 -4 = 0.0625 .

Continuing the conversation about the power of a number, it is logical to figure out how to find the value of the power. This process is called exponentiation. In this article we will study how exponentiation is performed, while we will touch on all possible exponents - natural, integer, rational and irrational. And according to tradition, we will consider in detail solutions to examples of raising numbers to various powers.

Page navigation.

What does "exponentiation" mean?

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

Definition.

Exponentiation- this is finding the value of the power of a number.

Thus, finding the value of the power of a number a with exponent r and raising the number a to the power r are 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 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 a number a to a fractional power m/n, first the nth root of the number a is taken, after which the resulting result is raised to an integer power m.

Let's look at solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We will show two solutions.

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

Second way. By the definition of a degree with a fractional exponent and based on the properties of the roots, the following equalities are true: . Now we extract the root , finally, we raise it to an integer power .

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

Answer:

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

Example.

Calculate (44.89) 2.5.

Solution.

Let us write the exponent in the form common fraction(if necessary, see the article): . Now we perform the 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 labor-intensive process (especially when the numerator and denominator fractional indicator the degrees are sufficient big numbers), which is usually carried out using computer technology.

To conclude this point, let us dwell on raising the number zero to a fractional power. Fractional degree zero of the form we gave the following meaning: when we have , and at zero to the m/n power is not defined. So, zero to a fractional positive power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the power of a number with an irrational exponent. At the same time, in practical purposes Usually it is enough to obtain the value of the degree up to a certain sign. Let us immediately note that this value in practice is calculated using electronic computer technology, since raising to ir rational degree manually requires large quantity cumbersome calculations. But still we will describe in general outline the essence of the action.

To obtain an approximate value of the power of a number a with an irrational exponent, some decimal approximation of the exponent is taken and the value of the power is calculated. This value is an approximate value of the power of the number a with an irrational exponent. The more accurate the decimal approximation of a number is taken initially, the more accurate the value of the degree will be obtained 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 irrational indicator: . Now we raise 2 to the rational power 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 the irrational exponent, for example, then we obtain a more accurate value of the original exponent: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics textbook for 5th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

The table of powers contains the values ​​of positive natural numbers from 1 to 10.

Entry 3 5 read “three to the fifth power.” In this notation, the number 3 is called the base of the power, the number 5 is the exponent, and the expression 3 5 is called the power.

To download the table of degrees, click on the thumbnail image.

Degree calculator

We invite you to try our powers calculator, which will help you raise any number to a power online.

Using the calculator is very simple - enter the number you want to raise to a power, then the number - the power and click on the "Calculate" button.

It is noteworthy that our online calculator powers can raise to both positive and negative powers. And for extracting roots there is another calculator on the site.

How to raise a number to a power.

Let's look at the process of exponentiation with an example. Suppose we need to raise the number 5 to the 3rd power. In the language of mathematics, 5 is the base, and 3 is the exponent (or simply the degree). And this can be written briefly as follows:

Exponentiation

And to find the value, we will need to multiply the number 5 by itself 3 times, i.e.

5 3 = 5 x 5 x 5 = 125

Accordingly, if we want to find the value of the number 7 to the 5th power, we must multiply the number 7 by itself 5 times, i.e. 7 x 7 x 7 x 7 x 7. Another thing is when you need to raise the number to a negative power.

How to raise to a negative power.

When raising to a negative power, you need to use a simple rule:

how to raise to a negative power

Everything is very simple - when raised to a negative power, we must divide one by the base to the power without the minus sign - that is, to the positive power. So to find the value

Table of powers of natural numbers from 1 to 25 in algebra

When solving various mathematical exercises, you often have to raise a number to a power, mainly from 1 to 10. And in order to quickly find these values, we have created a table of powers in algebra, which I will publish on this page.

First, let's look at the numbers from 1 to 6. The results here are not very large; you can check all of them on a regular calculator.

• 1 and 2 to the power of 1 to 10

Table of degrees

The power table is an indispensable tool when you need to raise a natural number within 10 to a power greater than two. It is enough to open the table and find the number opposite the desired base of the degree and in the column with the required degree - it will be the answer to the example. In addition to the convenient table, at the bottom of the page there are examples of raising natural numbers to powers up to 10. By selecting the required column with powers of the desired number, you can easily and simply find the solution, since all powers are arranged in ascending order.

Important nuance! The tables do not show raising to the zero power, since any number raised to the zero power is equal to one: a 0 =1

Multiplication tables, squares and powers

It's time to do a little math. Do you still remember how much it is if two are multiplied by two?

If anyone has forgotten, there will be four. It seems that everyone remembers and knows the multiplication table, however, I discovered great amount queries to Yandex like “multiplication table” or even “download multiplication table”(!). It is for this category of users, as well as for more advanced ones who are already interested in squares and powers, that I am posting all these tables. You can even download for your health! So:

10 to the 2nd degree + 11 to the 2nd degree + 12 to the 2nd degree + 13 to the 2nd degree + 14 to the second degree/365

Other questions from the category

Help me decide please)

Read also

solutions: 3x(to the 2nd power)-48= 3(X to the 2nd power)(x to the second power)-16)=(X-4)(X+4)

5) three point five. 6) nine point two hundred seven thousandths. 2) write down the number in the form of an ordinary fraction: 1)0.3. 2)0.516. 3)0.88. 4)0.01. 5)0.402. 5)0.038. 6)0.609. 7)0.91.8)0.5.9)0.171.10)0.815.11)0.27.12)0.081.13)0.803

What is 2 to the minus 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 powers?

What is 2 to the minus 1 power?

What is 2 to the minus 2 power?

What is 2 to the minus 3 power?

What is 2 to the minus 4th power?

What is 2 to the power of minus 5?

What is 2 to the minus 6th power?

What is 2 to the minus 7th power?

What is 2 to the power of minus 8?

What is 2 to the minus 9th power?

What is 2 to the power of minus 10?

The negative power of n ^(-a) can be expressed in following form 1/n^a.

2 to the power -1 = 1/2, if represented as a decimal fraction, then 0.5.

2 to the power - 2 = 1/4, or 0.25.

2 to the power -3= 1/8, or 0.125.

2 to the power -4 = 1/16, or 0.0625.

2 to the power -5 = 1/32, or 0.03125.

2 to the power - 6 = 1/64, or 0.015625.

2 to the power - 7 = 1/128, or 0.

2 to the power -8 = 1/256, or 0.

2 to the power -9 = 1/512, or 0.

2 to the power - 10 = 1/1024, or 0.

Similar calculations for other numbers can be found here: 3, 4, 5, 6, 7, 8, 9

The negative power of a number, at first glance, complex topic in algebra.

In fact, everything is very simple - mathematical calculations with the number "2" we carry out algebraic formula(see above), where instead of “a” we substitute the number “2”, and instead of “n” we substitute the power of the number. The calculator will help to significantly reduce the time in calculations.

Unfortunately, the site's text editor does not allow you to use mathematical symbols fractions and negative powers. Let's limit ourselves to capital alphanumeric information.

These are the simple numerical steps we ended up with.

A negative power of a number means that this number is multiplied by itself as many times as it is written in the power and then one is divided by the resulting number. For two:

• (-1) degree is 1/2=0.5;
• (-2) degree is 1/(2 2)=0.25;
• (-3) degree is 1/(2 2 2)=0.125;
• (-4) degree is 1/(2 2 2 2)=0.0625;
• (-5) degree is 1/(2 2 2 2 2)=0.03125;
• (-6) degree is 1/(2 2 2 2 2 2)=0.015625;
• (-7) degree is 1/(2 2 2 2 2 2 2)=0.078125;
• (-8) degree is 1/(2 2 2 2 2 2 2 2)=0,;
• (-9) degree is 1/(2 2 2 2 2 2 2 2 2)=0,;
• (-10) power is 1/(2 2 2 2 2 2 2 2 2 2)=0,.

Essentially, we simply divide each previous value by 2.

shkolnyie-zadachi.pp.ua

1) 33²: 11=(3*11)²: 11=3² * 11²: 11=9*11=99

2) 99²: 81=(9*11)²: 9²=9² * 11²: 9²=11²=121

The second degree means that the figure obtained during the calculations is multiplied by itself.

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Early spring, late spring, spring foliage, spring sun, spring day, spring has come, spring birds, cold spring, spring grass, spring breeze, spring rain, spring clothes, spring boots, spring is red, spring travel.

Question: 5*4 to the second power -(33 to the second power: 11) to the 2nd power: 81 SAY THE ANSWER BY ACTION

5*4 to the second power -(33 to the second power: 11) to the 2nd power: 81 SAY THE ANSWER BY ACTION

Answers:

5*4²-(33²: 11)²: 81= -41 1) 33²: 11=(3*11)²: 11=3² * 11²: 11=9*11=99 2) 99²: 81=(9* 11)²: 9²=9² * 11²: 9²=11²=121 3) 5*4²=5*16=80 4)= -41

5*4 (2) = 400 1) 5*4= 20 2) 20*20=:11(2)= 9 1) 33:11= 3 2) 3*3= 9 The second power means that the number that turned out to be multiplied by itself during calculations.

10 to the -2 power is how much.

1. 10 to the -2 power is the same as 1/10 to the 2 power, you square 10 and you get 1/100, which is equal to 0.01.

10^-2 = 1/10 * 1/10 = 1/(10*10) = 1/100 = 0.01

=) Dark you say? ..heh (from " White sun desert")

10 in -2 means 1 divided by 10 in 2. i.e. 0.01

0.01 Finished your studies!

10 to the 2nd power means 100

10 to the 1st power 10

if the degree is reduced by one, then the result decreases in this case by 10 times, therefore 10 to the power of 0 will be 1 (10/10)

10 to the power of -1 is 1/10

10 to the -2 power is 1/100 or 0.01

I didn’t understand what degree 2 or -2. if 2 before the answer is 100, if -2 then 0.01

100, it’s strange how you think it turns out to be 0.01.

This is 0.01 - I am responsible for the correctness!! ! And the fact that they wrote you 100 is if it were 10 to the 2nd power, so you don’t even have to doubt it

All this is ten to the minus second power

Is everything so difficult in the evening?

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 the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:

   • 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
     • 16 ∗ 4 = 64 (\displaystyle 16*4=64)
   • 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
     • 64 ∗ 4 = 256 (\displaystyle 64*4=256)
   • 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
     • 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
   • 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 improper fraction, then such a degree can be decomposed into two degrees 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 to the zero power is 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 .