There are many tables of degrees natural numbers. It is not possible to list them all. Here we give examples of some of these tables and tasks for finding values ​​from such tables.

Table of powers of the first natural numbers

Let's start with a table for finding the powers of natural numbers from $2$ to $12$ by powers from $1$ to $10$ (Table 1). Note that we do not give powers of $1$, because one will be equal to itself to any power.

It is necessary to find the values ​​​​from this table as follows: In the first column we find the number whose degree we are interested in. Remember the number of this line. Then, in the first term, we find the exponent and remember the found column. The intersection of the found row and column will give us the answer.

Example 1

Find $8^7$

We find the number $8$ in the first column: we get the 8th line.

We see that the number $2097152$ is at their intersection. Hence

Tables of powers of natural numbers from $1$ to $100$

Tables of degrees from $1$ to $100$ are also quite popular. It is impossible to give all of them, so we will give as an example such tables for squares and cubes of such natural numbers (Table 2 and Table 3).

These tables are reminiscent of the well-known multiplication tables, so we think the reader will not find it difficult to use these tables.

Example 2

a) Given value we find in table $2$ in $8$ plate:

b) We find this value in table $3$ in $3$ plate:

Table of squares of natural numbers from $10$ to $99$

Another popular table is the table of squares of numbers from $10$ to $99$ (table 4), that is, all decimal numbers.

It is necessary to find the values ​​​​from this table as follows: In the first column we find the number of tens of the number of interest to us. Remember the number of this line. Then, in the first term, we find the number of units of the number of interest and remember the found column. The intersection of the found row and column will give us the answer.

Example 3

Find $37^2$

We find the number $3$ in the first column: we get the 4th line.

We find the number $7$ in the first row: we get the 8th column.

We see that at their intersection is the number $1369$. Hence

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 we will touch 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.

Decision.

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 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 or mixed number, in these cases it should be replaced by the corresponding ordinary fraction, after which exponentiation should be performed.

Example.

Calculate (44.89) 2.5 .

Decision.

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 fractional indicator degrees are sufficiently large numbers), which is usually carried out using computer technology.

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 in fractional positive degree equals zero, for example, . And zero in fractional negative degree 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 raising to ir rational degree manually requires a large number cumbersome calculations. However, we will describe in general terms essence of action.

To get an approximate value of the power of a number a with ir rational indicator, 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 we 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 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 successful delivery OGE or USE 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 like addition, subtraction, multiplication or division.

Now I'll explain everything human language very simple examples. Pay attention. Examples are elementary, but explain important things.

Let's start with addition.

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

Now multiplication.

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

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

Here is the multiplication table. Repeat.

And another, prettier one:

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

Raising a number to a power

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

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

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

Real life example #1

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

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

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

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

Real life example #2

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

Real life example #3

Now the cube or the third power of a number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. 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 natural indicator

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

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

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

Is there some more 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).

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

Definition. Raise a number to natural degree means to multiply a number by itself times:
.

Degree properties

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

Let's see what is and ?

A-priory:

How many multipliers are there in total?

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

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

Example: Simplify the expression.

Decision:

Example: Simplify the expression.

Decision: It is important to note that in our rule necessarily must be 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 which signs ("" or "") will have degrees of positive and negative numbers?

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

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

Determine for yourself what sign the following expressions will have:

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.

whole positive number , 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 exponent is, let's do as in last time: 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 independent decision:

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 a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But here a problem arises.

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

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

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

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

So if:

• - natural number;
• is an integer;

Examples:

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

5 practice examples

Analysis of 5 examples for training

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

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

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

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

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

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

...degree with integer negative indicator - 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, 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 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:

AT this case,

It turns out that:

Answer: .

2. We give fractions in exponents of k the same kind: Either both decimals or both normal. We get, for example:

Answer: 16

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

ADVANCED LEVEL

Definition of degree

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

• — base of degree;
• - exponent.

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

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

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

If the exponent is positive integer number:

erection to zero power:

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

If the exponent is integer negative number:

(because it is impossible to divide).

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

Examples:

Degree with rational exponent

• - natural number;
• is an integer;

Examples:

Degree properties

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

Let's see: what is and?

A-priory:

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

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

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

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

One more important note: this rule is - only for products of powers!

Under no circumstances should I write that.

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

Let's rearrange it like this:

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

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

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

Power with a negative base.

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

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

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

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

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

And so on ad infinitum: with each subsequent multiplication, the sign will change. It is possible to formulate such simple rules:

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

Determine for yourself what sign the following expressions will have:

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 does not 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 the degrees and divide them into each other, divide them into pairs and get:

Before disassembling last rule Let's take a look at a few examples.

Calculate the values ​​of expressions:

Solutions :

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

We get:

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

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

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

Let's go back to the example:

And again the formula:

So now the last rule:

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

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

Example:

Degree with irrational exponent

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

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

1)

2)

3)

Answers:

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

SECTION SUMMARY AND BASIC FORMULA

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

Degree with integer exponent

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

Degree with rational exponent

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

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree properties

Features of degrees.

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

NOW YOU HAVE A WORD...

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

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

Let's consider a sequence of numbers, the first of which is 1, and each subsequent one is twice as large: 1, 2, 4, 8, 16, ... Using exponents, it can be written in the equivalent form: 2 0 , 2 1 , 2 2 . 2 3 , 2 4 , ... It is called quite expectedly: a sequence of powers of two. It would seem that there is nothing outstanding in it - a sequence as a sequence, no better and no worse than others. However, it has some very remarkable properties.

Undoubtedly, many readers have met her in classical history about the inventor of chess, who asked the ruler as a reward for the first cell of the chessboard one grain of wheat, for the second - two, for the third - four, and so on, doubling the number of grains all the time. It is clear that their total number is equal to

S= 2 0 + 2 1 + 2 2 + 2 3 + 2 4 + ... + 2 63 . (1)

But since this amount is incredibly large and many times exceeds the annual grain harvest around the world, it turned out that the sage skinned the ruler like sticky.

However, let us now ask ourselves another question: how to calculate the value of S? Owners of a calculator (or, moreover, a computer) can easily perform multiplications in a foreseeable time, and then add the resulting 64 numbers, receiving the answer: 18,446,744,073,709,551,615. And since the amount of calculations is considerable, the probability of error is very high.

Who is more cunning can see in this sequence geometric progression. Those who are not familiar with this concept (or those who simply forgot standard formula amounts geometric progression) can use the following reasoning. Let's multiply both sides of equality (1) by 2. Since when doubling the power of two, its exponent increases by 1, we get

2S = 2 1 + 2 2 + 2 3 + 2 4 + ... + 2 64 . (2)

Now from (2) subtract (1). On the left side, of course, it turns out 2 S – S = S. On the right side, there will be a massive mutual destruction of almost all powers of two - from 2 1 to 2 63 inclusive, and only 2 64 - 2 0 \u003d 2 64 - 1 will remain. So:

S= 2 64 – 1.

Well, the expression has been noticeably simplified, and now, having a calculator that allows you to raise to a power, you can find the value of this quantity without the slightest problem.

And if there is no calculator - what to do? Multiply in a column of 64 deuces? What else was missing! An experienced engineer or applied mathematician for whom main factor- time, could quickly estimate response, i.e. find it approximately with acceptable accuracy. As a rule, in everyday life (and in most natural sciences) an error of 2–3% is quite acceptable, and if it does not exceed 1%, then this is just great! It turns out that it is possible to calculate our grains with such an error without a calculator at all, and in just a few minutes. How? Now you will see.

So, it is necessary to find the product of 64 twos as accurately as possible (we will immediately discard the unit due to its insignificance). Let's break them into a separate group of 4 twos and another 6 groups of 10 twos. The product of twos in separate group equals 2 4 = 16. And the product of 10 twos in each of the other groups is 2 10 = 1024 (be sure who doubts!). But 1024 is about 1000, i.e. 10 3 . So S should be close to the product of the number 16 by 6 numbers, each of which is equal to 10 3 , i.e. S ≈ 16 10 18 (because 18 = 3 6). True, the error here is still quite large: after all, 6 times when replacing 1024 by 1000, we were mistaken by 1.024 times, and in total we were mistaken, as it is easy to see, by 1.024 6 times. So now - additionally multiply 1.024 six times by itself? No, let's go! It is known that for the number X, which is many times less than 1, with high precision the following approximate formula is valid: (1 + x) n ≈ 1 + xn.

Therefore 1.024 6 = (1 + 0.24) 6 ≈ 1 + 0.24 6 = 1.144. Therefore, we need to multiply the number 16 10 18 found by us by the number 1.144, resulting in 18,304,000,000,000,000,000, and this differs from the correct answer by less than 1%. What we were looking for!

In this case, we were very lucky: one of the powers of two (namely, the tenth) turned out to be very close to one of the powers of ten (namely, the third). This allows us to quickly evaluate the value of any power of two, not necessarily the 64th. Among the powers of other numbers, this is not common. For example, 5 10 differs from 10 7 also by 1.024 times, but ... in a smaller direction. However, this is a berry of the same field: since 2 10 5 10 \u003d 10 10, then how many times 2 10 surpasses 10 3 , the same number of times 5 10 smaller than 10 7 .

Other interesting feature of the sequence under consideration is that any natural number can be constructed from various powers of two, and the only way. For example, for the number current year we have

2012 = 2 2 + 2 3 + 2 4 + 2 6 + 2 7 + 2 8 + 2 9 + 2 10 .

It is not difficult to prove this possibility and uniqueness. Let's start with opportunities. Suppose we need to represent in the form of a sum various degrees two is some natural number N. First, we write it as a sum N units. Since the unit is 2 0, then initially N there is a sum identical powers of two. Then we'll start pairing them up. The sum of two numbers equal to 2 0 is 2 1 , so the result is obviously less the number of terms equal to 2 1 , and, possibly, one number 2 0 if it did not find a pair. Next, we combine the same terms 2 1 in pairs, getting an even smaller number of numbers 2 2 (here, the appearance of an unpaired power of two 2 1 is also possible). Then we again combine equal terms in pairs, and so on. Sooner or later, the process will end, because the number of identical powers of two decreases after each union. When it becomes equal to 1 - it's over. It remains to add up all the resulting unpaired powers of two - and the representation is ready.

As for proof uniqueness representations, then the method "by contradiction" is well suited here. Let the same number N was able to present in the form two sets of different powers of 2 that do not exactly match (i.e., there are powers of 2 that are in one set but not in another, and vice versa). First, let's discard all matching powers of two from both sets (if any). You get two representations of the same number (less than or equal to N) as a sum of different powers of two, and all degrees in submissions different. In each of the representations, select greatest degree. By virtue of the above, for two representations these degrees different. The representation for which this degree is greater is called first, other - second. So, let in the first representation, the largest power is 2 m, then in the second it obviously does not exceed 2 m-one . But since (and we have already encountered this above, counting the grains on the chessboard), the equality

2m = (2m –1 + 2m –2 + ... + 2 0) + 1,

then 2 m strictly more sums of all powers of two not exceeding 2 m-one . For this reason, the largest power of two included in the first representation is probably greater than the sum all powers of two included in the second representation. Contradiction!

In fact, we have just justified the possibility of writing numbers in binary number system. As you know, it uses only two digits - zero and one, and each natural number is written in the binary system in a unique way (for example, the 2012 mentioned above - as 11 111 011 100). If we number the digits (binary digits) from right to left, starting from zero, then the numbers of those digits in which there are units will just be the exponents of the twos included in the representation.

Less known next property sets of integer non-negative powers of two. Let's arbitrarily assign a minus sign to some of them, that is, from the positive ones we will make them negative. The only requirement is that the result of both positive and negative numbers is an infinite number. For example, you can assign a minus sign to every fifth power of two, or, say, leave positive only the numbers 2 10 , 2 100 , 2 1000 , and so on - there are as many options as you like.

Surprisingly, any whole the number can be (and, moreover, in a unique way) represented as the sum of various terms of our "positive-negative" sequence. And it is not very difficult to prove this (for example, by induction on exponents of twos). main idea evidence - the presence of arbitrarily large absolute value both positive and negative terms. Try doing the proof yourself.

It is interesting to observe the last digits of the members of the sequence of powers of two. Since each subsequent number in the sequence is obtained by doubling the previous one, the last digit of each of them is completely determined by the last digit previous date. And since various numbers a limited number, the sequence of last digits of powers of two is simply obliged be periodic! The length of the period, of course, does not exceed 10 (since that is how many digits we use), but this is a very overestimated value. Let's try to evaluate it without writing out the sequence itself yet. It is clear that the last digits of all powers of two, starting from 2 1 , even. In addition, zero cannot be among them - because a number ending in zero is divisible by 5, which cannot be suspected of a power of two. And since there are only four even digits without zero, the length of the period does not exceed 4.

Verification shows that this is the case, and the periodicity appears almost immediately: 1, 2, 4, 8, 6, 2, 4, 8, 6, ... - in full accordance with the theory!

No less successfully can one estimate the length of the period of the last pair of digits in a sequence of powers of two. Since all powers of two, starting from 2 2 , are divisible by 4, the numbers formed by their last two digits are also divisible by 4. No more than two-digit numbers, divisible by 4, there are only 25 (for single-digit numbers, we consider zero as the penultimate digit), but five numbers ending in zero must be thrown out of them: 00, 20, 40, 60 and 80. So the period can contain no more than 25 - 5 = 20 numbers. The check shows that it is, the period begins with the number 2 2 and contains pairs of numbers: 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72 , 44, 88, 76, 52, and then again 04 and so on.

Similarly, one can prove that the length of the period of the last m digits of a sequence of powers of two does not exceed 4 5 m–1 (moreover, in fact, she is equal to 4 5 m–1 , but this is much more difficult to prove).

So, quite severe restrictions are imposed on the last digits of the powers of two. and how about first numbers? Here the situation is almost the opposite. It turns out that for any set of digits (the first of which is not zero) there is a power of two starting with this set of digits. And such powers of two infinitely many! For example, there are an infinite number of powers of two starting with the digits 2012 or, say, 3,333,333,333,333,333,333,333.

And if we consider only one very first digit of various powers of two - what values ​​\u200b\u200bcan it take? It is easy to make sure that any - from 1 to 9 inclusive (of course, there is no zero among them). But which ones are more common and which ones are less common? Somehow, it is not immediately clear why one number should occur more often than another. However, deeper reflections show that just the same occurrence of numbers is not to be expected. Indeed, if the first digit of any power of two is 5, 6, 7, 8 or 9, then the first digit of the power of two following it will be necessarily unit! Therefore, there must be a "skew", at least towards unity. Therefore, it is unlikely that the rest of the figures will be "equally represented".

Practice (namely, direct computer calculation for the first few tens of thousands of powers of two) confirms our suspicions. Here is the relative proportion of the first digits of powers of two, rounded to 4 decimal places:

1 - 0,3010
2 - 0,1761
3 - 0,1249
4 - 0,0969
5 - 0,0792
6 - 0,0669
7 - 0,0580
8 - 0,0512
9 - 0,0458

As you can see, this value decreases with the growth of digits (and therefore the same unit is about 6.5 times more likely to be the first digit of powers of two than nine). Strange as it may seem, but practically the same ratio of the number of first digits will take place for almost any sequence of degrees - not only two, but, say, three, five, eight, and in general almost any numbers, including non-integer ones (the only exceptions are some "special" numbers). The reasons for this are very deep and complex, and to understand them one must know the logarithms. For those who are familiar with them, let's lift the veil: it turns out that the relative fraction of powers of two, decimal notation that start with a number F(for F= 1, 2, ..., 9) is lg ( F+ 1) – lg ( F), where lg is the so-called decimal logarithm, equal to the exponent to which the number 10 must be raised to get the number under the sign of the logarithm.

Using the connection between the powers of two and five mentioned above, A. Kanel discovered an interesting phenomenon. Let's select a few digits from the sequence of the first digits of powers of two (1, 2, 4, 8, 1, 3, 6, 1, 2, 5, ...) contract and write them in reverse order. It turns out that these numbers will certainly meet also in a row, starting from some place, in the sequence of the first digits of powers of five.

Powers of two are also a kind of "generator" for the production of well-known perfect numbers, which is equal to the sum of all its divisors, excluding itself. For example, the number 6 has four divisors: 1, 2, 3 and 6. Let's discard the one that is equal to the number 6 itself. There are three divisors left, the sum of which is exactly equal to 1 + 2 + 3 = 6. Therefore, 6 is a perfect number.

To get a perfect number, take two consecutive powers of two: 2 n-1 and 2 n. Decrease the largest of them by 1, we get 2 n– 1. It turns out that if this is a prime number, then multiplying it by the previous power of two, we form a perfect number 2 n –1 (2n- one). For example, when P= 3 we get the original numbers 4 and 8. Since 8 - 1 = 7 is a prime number, then 4 7 = 28 is a perfect number. Moreover, at one time Leonhard Euler proved that all even perfect numbers look like this. Odd perfect numbers have not yet been discovered (and few people believe in their existence).

Powers of two are closely related to the so-called Catalan numbers, whose sequence has the form 1, 1, 2, 5, 14, 42, 132, 429... They often arise when solving various combinatorial problems. For example, in how many ways can a convex n-gon into triangles with non-intersecting diagonals? All the same Euler found out that this value is equal to ( n- 1)th number of Catalan (we denote it K n-1), and he found that K n = K n-fourteen n – 6)/n. The Catalan number sequence has many interesting properties, and one of them (just related to the topic of this article) is that sequence numbers all odd Catalan numbers are powers of two!

Powers of two are often found in various problems, not only in conditions, but also in answers. Take, for example, the once popular (and still not forgotten) tower of hanoi. This was the name of a puzzle game invented in the 19th century. French mathematician E. Luca. It contains three rods, one of which is worn n discs with a hole in the middle of each. The diameters of all disks are different, and they are arranged in descending order from bottom to top, i.e. the largest disk is at the bottom (see figure). It turned out like a tower of disks.

It is required to transfer this tower to another rod, observing the following rules: shift the disks exactly one at a time (removing the upper disk from any rod) and always put only the smaller disk on the larger one, but not vice versa. The question is: what is the minimum number of moves required for this? (We call a move the removal of a disk from one rod and putting it on another.) Answer: it is equal to 2 n– 1, which is easily proved by induction.

Let for n disks, the required minimum number of moves is X n. Let's find X n+1 . In the process of work, sooner or later it will be necessary to remove the largest disk from the rod, on which all the disks were originally put on. Since this disk can only be put on an empty rod (otherwise it will “press down” a smaller disk, which is prohibited), then all the upper n disks will have to be transferred to the third rod first. This will require no less X n moves. Next, we transfer the largest disk to an empty rod - here's another move. Finally, in order to “squeeze” it from above with smaller n disks, again it will take no less X n moves. So, X n +1 ≥Xn + 1 +Xn = 2X n+ 1. On the other hand, the actions described above show how you can cope with the task exactly 2 X n+ 1 moves. Therefore, finally X n +1 =2X n+ 1. Received recurrence relation, but in order to bring it to a "normal" form, we must also find X one . Well, it's as simple as that: X 1 = 1 (there simply can't be less!). It is not difficult, based on these data, to find out that X n = 2n– 1.

Here is another interesting challenge:

Find all natural numbers that cannot be represented as the sum of several (at least two) consecutive natural numbers.

Let's check first smallest numbers. It is clear that the number 1 in specified form unimaginable. But all odd ones that are greater than 1 can, of course, be represented. Indeed, any even number greater than 1 can be written as 2 k + 1 (k- natural), which is the sum of two consecutive natural numbers: 2 k + 1 = k + (k + 1).

What about even numbers? It is easy to see that the numbers 2 and 4 cannot be represented in the required form. Maybe it's the same for all even numbers? Alas, the next even number refutes our assumption: 6 \u003d 1 + 2 + 3. But the number 8 again does not lend itself. Truth, next numbers again yield to the onslaught: 10 = 1 + 2 + 3 + 4, 12 = 3 + 4 + 5, 14 = 2 + 3 + 4 + 5, but 16 is again unimaginable.

Well, the accumulated information allows us to draw preliminary conclusions. Please note: could not be presented in the specified form only powers of two. Is this true for the rest of the numbers? It turns out yes! Indeed, consider the sum of all natural numbers from m before n inclusive. Since there are at least two of them in total, then n > m. As is well known, the sum of successive terms arithmetic progression(and this is what we are dealing with!) is equal to the product of the half-sum of the first and last terms and their number. The half sum is ( n + m)/2, and the number of numbers is n – m+ 1. Therefore, the sum is ( n + m)(n – m+ 1)/2. Note that the numerator contains two factors, each of which strictly more 1, and their parity is different. It turns out that the sum of all natural numbers from m before n inclusive is divisible by an odd number greater than 1, and therefore cannot be a power of two. So now it is clear why it was not possible to represent powers of two in the right form.

It remains to make sure that not a power of two can be imagined. As for odd numbers, we have already dealt with them above. Take any even number that is not a power of two. Let the largest power of 2 it is divisible by be 2 a (a- natural). Then if the number is divided by 2 a, it will already odd a number greater than 1, which we will write in a familiar form - as 2 k+ 1 (k- also natural). So, in general, our even number, which is not a power of two, is 2 a (2k+ 1). Now let's look at two options:

1. 2 a+1 > 2k+ 1. Take the sum 2 k+ 1 consecutive natural numbers, the average of which is equal to 2 a. It is easy to see that then least of which is equal to 2 a-k, and the largest is 2 a + k, and the smallest (and, hence, all the others) is positive, i.e., really natural. Well, the sum, obviously, is just 2 a(2k + 1).
2. 2 a+1 < 2k+ 1. Take the sum 2 a+1 consecutive natural numbers. Can't be specified here. the average number, because the number of numbers is even, but indicate a couple of medium numbers you can: let them be numbers k and k+ 1. Then least of all numbers is k+ 1 – 2a(and also positive!) and the largest is equal to k+ 2a. Their sum is also 2 a(2k + 1).

That's all. So, the answer is: non-representable numbers are powers of two, and only they.

And here is another problem (it was first proposed by V. Proizvolov, but in a slightly different formulation):

The garden plot is surrounded by a solid fence of N planks. By order of Aunt Polly, Tom Sawyer whitewashes the fence, but own system: moving clockwise all the time, first whitens an arbitrary board, then skips one board and whitens the next, then skips two boards and whitens the next, then skips three boards and whitens the next, and so on, each time skipping one more board (with some boards can be whitewashed several times - this does not bother Tom).

Tom thinks that under such a scheme, sooner or later all the boards will be whitewashed, and Aunt Polly is sure that at least one board will remain unwhitewashed, no matter how much Tom works. Under what N is Tom right, and under what is Aunt Polly?

The described system of whitewashing seems rather chaotic, so it may initially seem that for any (or almost any) N each board will someday get its share of lime, i.e., primarily, right Tom. But the first impression is deceptive, because in fact Tom is only right for the values N, which are powers of two. For others N there is a board that will remain forever unwhitewashed. The proof of this fact is rather cumbersome (although, in principle, it is not difficult). We invite the reader to do it himself.

That's what they are - powers of two. It looks simpler than simple, but as you dig ... And we have touched here not all the amazing and mysterious properties of this sequence, but only those that caught our eye. Well, the reader is given the right to independently continue research in this area. No doubt they will be fruitful.

Zero number).
And not only deuces, as noted earlier!
Thirsty for details, you can read the article by V. Boltyansky “Do powers of two often begin with one?” (“Quantum” No. 5, 1978), as well as an article by V. Arnold “Statistics of the first digits of powers of two and the redivision of the world” (“Quantum,” No. 1, 1998).
See problem M1599 from the "Kvant" problem book ("Kvant" No. 6 for 1997).
Currently, 43 perfect numbers are known, the largest of which is 2 30402456 (2 30402457 - 1). It contains over 18 million digits.

Table of powers 2 (twos) from 0 to 32

The above table, in addition to the power of two, shows the maximum numbers that a computer can store for a given number of bits. And both for integers and numbers with a sign.

Historically, computers used the binary number system, and, accordingly, data storage. Thus, any number can be represented as a sequence of zeros and ones (bits of information). There are several ways to represent numbers as a binary sequence.

Consider the simplest of them - this is a positive integer. Then what more number we need to write, the longer the sequence of bits we need.

Below is table of powers of number 2. It will give us a representation of the required number of bits that we need to store numbers.

How to use table of powers of two?

The first column is power of two, which simultaneously denotes the number of bits that represents the number.

Second column - value twos to the corresponding power (n).

An example of finding the power of a number 2. We find the number 7 in the first column. We look along the line to the right and find the value two to the seventh power(2 7 ) is 128

Third column - the maximum number that can be represented with a given number of bits(in the first column).

Example of determining the maximum unsigned integer. Using the data from the previous example, we know that 2 7 = 128 . This is true if we want to understand what amount of numbers, can be represented using seven bits. But since the first number is zero, then the maximum number that can be represented using seven bits is 128 - 1 = 127 . This is the value of the third column.

Power of two (n)	Power of two value 2n	Maximum unsigned number, written with n bits	Maximum signed number, written with n bits
0	1	-	-
1	2	1	-
2	4	3	1
3	8	7	3
4	16	15	7
5	32	31	15
6	64	63	31
7	128	127	63
8	256	255	127
9	512	511	255
10	1 024	1 023	511
11	2 048	2 047	1023
12	40 96	4 095	2047
13	8 192	8 191	4095
14	16 384	16 383	8191
15	32 768	32 767	16383
16	65 536	65 535	32767
17	131 072	131 071	65 535
18	262 144	262 143	131 071
19	524 288	524 287	262 143
20	1 048 576	1 048 575	524 287
21	2 097 152	2 097 151	1 048 575
22	4 194 304	4 194 303	2 097 151
23	8 388 608	8 388 607	4 194 303
24	16 777 216	16 777 215	8 388 607
25	33 554 432	33 554 431	16 777 215
26	67 108 864	67 108 863	33 554 431
27	134 217 728	134 217 727	67 108 863
28	268 435 456	268 435 455	134 217 727
29	536 870 912	536 870 911	268 435 455
30	1 073 741 824	1 073 741 823	536 870 911
31	2 147 483 648	2 147 483 647	1 073 741 823
32	4 294 967 296	4 294 967 295	2 147 483 647