To what extent does 2 give. Learn more about degrees and exponentiation. Calculate expressions

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: 92067

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 .

Table of powers of numbers from 1 to 10. Online powers calculator. Interactive table and images of the table of degrees in high quality.

Degree calculator

Number

Degree

Calculate Clear

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With this calculator you can calculate the power of any natural number online. Enter the number, degree and click the “calculate” button.

Table of degrees from 1 to 10

n	1	2	3	4	5	6	7	8	9	10
1n	1	1	1	1	1	1	1	1	1	1
2n	2	4	8	16	32	64	128	256	512	1024
3n	3	9	27	81	243	729	2187	6561	19683	59049
4n	4	16	64	256	1024	4096	16384	65536	262144	1048576
5n	5	25	125	625	3125	15625	78125	390625	1953125	9765625
6n	6	36	216	1296	7776	46656	279936	1679616	10077696	60466176
7n	7	49	343	2401	16807	117649	823543	5764801	40353607	282475249
8n	8	64	512	4096	32768	262144	2097152	16777216	134217728	1073741824
9n	9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401
10n	10	100	1000	10000	100000	1000000	10000000	100000000	1000000000	10000000000

Table of degrees from 1 to 10

1 1 = 1 1 2 = 1 1 3 = 1 1 4 = 1 1 5 = 1 1 6 = 1 1 7 = 1 1 8 = 1 1 9 = 1 1 10 = 1	2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 2 6 = 64 2 7 = 128 2 8 = 256 2 9 = 512 2 10 = 1024	3 1 = 3 3 2 = 9 3 3 = 27 3 4 = 81 3 5 = 243 3 6 = 729 3 7 = 2187 3 8 = 6561 3 9 = 19683 3 10 = 59049	4 1 = 4 4 2 = 16 4 3 = 64 4 4 = 256 4 5 = 1024 4 6 = 4096 4 7 = 16384 4 8 = 65536 4 9 = 262144 4 10 = 1048576	5 1 = 5 5 2 = 25 5 3 = 125 5 4 = 625 5 5 = 3125 5 6 = 15625 5 7 = 78125 5 8 = 390625 5 9 = 1953125 5 10 = 9765625
6 1 = 6 6 2 = 36 6 3 = 216 6 4 = 1296 6 5 = 7776 6 6 = 46656 6 7 = 279936 6 8 = 1679616 6 9 = 10077696 6 10 = 60466176	7 1 = 7 7 2 = 49 7 3 = 343 7 4 = 2401 7 5 = 16807 7 6 = 117649 7 7 = 823543 7 8 = 5764801 7 9 = 40353607 7 10 = 282475249	8 1 = 8 8 2 = 64 8 3 = 512 8 4 = 4096 8 5 = 32768 8 6 = 262144 8 7 = 2097152 8 8 = 16777216 8 9 = 134217728 8 10 = 1073741824	9 1 = 9 9 2 = 81 9 3 = 729 9 4 = 6561 9 5 = 59049 9 6 = 531441 9 7 = 4782969 9 8 = 43046721 9 9 = 387420489 9 10 = 3486784401	10 1 = 10 10 2 = 100 10 3 = 1000 10 4 = 10000 10 5 = 100000 10 6 = 1000000 10 7 = 10000000 10 8 = 100000000 10 9 = 1000000000 10 10 = 10000000000

Theory

Degree of is an abbreviated form of the operation of repeatedly multiplying a number by itself. The number itself in in this case called - degree basis, and the number of multiplication operations is exponent.

a n = a×a ... ×a

the entry reads: "a" to the power of "n".

"a" is the base of the degree

"n" - exponent

4 6 = 4 × 4 × 4 × 4 × 4 × 4 = 4096

This expression reads: 4 to the power of 6 or the sixth power of the number four or raise the number four to the sixth power.

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Let's consider a sequence of numbers, the first of which is equal to 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: sequence of powers of two. It would seem that there is nothing outstanding in it - consistency is like consistency, no better and no worse than others. However, it has very remarkable properties.

Undoubtedly, many readers have encountered it in the classic story about the inventor of chess, who asked the ruler as a reward for the first square of the chessboard one grain of wheat, for the second - two, for the third - four, and so on, all the time doubling the number of grains. 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 fleeced the ruler like a stick.

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

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

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

Now from (2) we subtract (1). On the left side, of course, it turns out to be 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 = 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 you don’t have a calculator, what should you do? Multiply 64 twos into a column? What else was missing! An experienced engineer or applied mathematician, for whom time is the main factor, would be able to quickly estimate answer, 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 that’s just great! It turns out that you can calculate our grains with such an error without a calculator at all, and in just a few minutes. How? You'll see now.

So, we need to find the product of 64 twos as accurately as possible (we will immediately discard the one due to its insignificance). Let's divide them into a separate group of 4 twos and another 6 groups of 10 twos. The product of twos in a separate group is equal to 2 4 = 16. And the product of 10 twos in each of the other groups is equal to 2 10 = 1024 (see if you doubt it!). But 1024 is about 1000, i.e. 10 3. That's why 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 (since 18 = 3·6). True, the error here is still large: after all, 6 times when replacing 1024 by 1000 we were mistaken 1.024 times, and in total we were mistaken, as is easy to see, 1.024 6 times. So what now - additionally multiply 1.024 six times by itself? No, we'll get by! It is known that for the number X, which is many times less than 1, the following approximate formula is valid with high accuracy: (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 we found 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%. That's what we wanted!

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 rare. For example, 5 10 differs from 10 7 also by 1.024 times, but... to a lesser extent. However, this is the same thing: since 2 10 5 10 = 10 10, then how many times 2 10 superior 10 3, the same number of times 5 10 less, than 10 7 .

Other interesting feature 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 current year number we have

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

Proving this possibility and uniqueness is not difficult. Let's start with possibilities. Suppose we need to represent a certain natural number as a sum of different powers of two N. First, let's write it as a sum N units. Since one is 2 0, then initially N there is a sum identical powers of two. Then we will start combining them in pairs. 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 no pair was found for it. Next, we combine identical terms 2 1 in pairs, obtaining an even smaller number of numbers 2 2 (here, too, the appearance of an unpaired power of two 2 1 is possible). Then we again combine equal terms in pairs, and so on. Sooner or later the process will end, because the amount equal degrees the number of twos decreases after each union. When it becomes equal to 1, the matter is over. All that remains is to add up all the resulting unpaired powers of two - and the performance is ready.

As for the proof uniqueness representations, then the “by contradiction” method is well suited here. Let the same number N was able to be represented in the form two sets of different powers of two that do not completely coincide (that is, there are powers of two that are included 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 will get two representations of the same number (less than or equal to N) as a sum of various powers of two, and All degrees in representations different. In each of the representations we highlight the greatest degree. Due to the above, for two representations these degrees different. We call the representation for which this degree is greater first, other - second. So, let in the first representation the greatest degree be 2 m, then in the second it obviously does not exceed 2 m-1 . But since (and we have already encountered this above, counting the grains on the chessboard) the equality is true

2m = (2m –1 + 2m –2 + ... + 2 0) + 1,

then 2 m strictly more the sum of all powers of 2 not exceeding 2 m-1 . For this reason, the largest power of two included in the first representation is certainly greater than the sum everyone 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 above-mentioned 2012 - 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 ones will precisely be indicators of the powers of twos included in the representation.

Less well known is the following property of the set of integer non-negative powers of two. Let's arbitrarily assign a minus sign to some of them, i.e., turn positive ones into negative ones. The only requirement is that the result of both positive and negative numbers be an infinite number. For example, you can assign a minus sign to every fifth power of two or, for example, leave 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 (and in the only way) be represented as the sum of the various terms of our “positive-negative” sequence. And it is not very difficult to prove this (for example, by induction on exponents of powers of twos). main idea evidence - the presence of arbitrarily large absolute value both positive and negative terms. Try the proof yourself.

It is interesting to observe the last digits of the terms 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 of the previous number. And since there are a limited number of different digits, the sequence of the last digits of powers of two is simply obliged be periodic! The length of the period, naturally, does not exceed 10 (since that is how many numbers we use), but this is a greatly overestimated value. Let's try to evaluate it without writing out the sequence itself for now. It is clear that the last digits of all powers of two, starting with 2 1, even. In addition, there cannot be a zero among them - because a number ending in zero is divisible by 5, which cannot be suspected of being a power of two. And since there are only four even digits without zero, the length of the period does not exceed 4.

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

It is no less successful to estimate the length of the period of the last pair of digits of a sequence of powers of two. Since all powers of two, starting with 2 2, are divisible by 4, then the numbers formed by their last two digits are divisible by 4. At most double digit numbers There are only 25 divisible by 4 (for single-digit numbers we consider zero to be the penultimate digit), but from them we must eliminate five numbers ending in zero: 00, 20, 40, 60 and 80. So the period can contain no more than 25 - 5 = 20 numbers. Checking shows that this is the case, 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, it can be proven that the length of the period of the last m the digits of the sequence of powers of two do not exceed 4 5 m–1 (moreover, in fact she equal to 4·5 m–1, but this is much more difficult to prove).

So, quite strict restrictions are imposed on the last digits of the powers of two. What 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 ​​can it take? It is easy to verify that any are from 1 to 9 inclusive (of course, there is no zero among them). But which of them are more common and which are less common? Somehow, it is not immediately obvious why one number should occur more often than another. However, deeper reflections show that exactly equal occurrence of numbers cannot 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 next power of two will necessarily be unit! Therefore, there must be a “skew” at least towards unity. Therefore, it is unlikely that the remaining numbers will be “equally represented.”

Practice (namely, direct computer calculations for the first several tens of thousands of powers of two) confirms our suspicions. Here is the relative proportion of the first digits of the 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 we see, as the numbers increase, this value decreases (and therefore the same unit is approximately 6.5 times more likely to be the first digit of powers of two than nine). Strange as it may seem, almost the same ratio of the numbers of the first digits will occur for almost any sequence of degrees - not only two, but, say, three, five, eight and in general almost anyone 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 you need to know logarithms. For those who are familiar with them, let us lift the veil: it turns out that the relative proportion of powers of two, decimal notation which starts with a number F(For F= 1, 2, ..., 9), is log ( F+ 1) – log ( F), where lg is the so-called decimal logarithm, equal to the exponent to which the number 10 must be raised to obtain the number under the logarithm sign.

Using the above-mentioned connection between the powers of two and five, A. Canel discovered an interesting phenomenon. Let's select several numbers 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 a certain place, in the sequence of the first digits of the powers of five.

Powers of two are also a kind of “generator” for the production of well-known perfect numbers, which are equal to the sum of all their 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. Three divisors remain, the sum of which is exactly 1 + 2 + 3 = 6. Therefore, 6 is a perfect number.

To obtain a perfect number, take two successive powers of two: 2 n–1 and 2 n. Reduce the largest of them by 1, we get 2 n– 1. It turns out that if this is a prime number, then by multiplying it by the previous power of two, we form the perfect number 2 n –1 (2n- 1). 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 Leonard Euler proved that everything even perfect numbers have exactly this form. 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, the sequence of which is 1, 1, 2, 5, 14, 42, 132, 429... They often arise when solving various combinatorial problems. For example, in how many ways can you split a convex n-gon into triangles with disjoint diagonals? The same Euler found out that this value is equal to ( n– 1) to the Catalan number (we denote it K n–1), and he also found out that K n = K n-14 n – 6)/n. The sequence of Catalan numbers has many interesting properties, and one of them (just related to the topic of this article) is that the ordinal numbers of all odd Catalan numbers are powers of two!

Powers of two are often found in various problems, not only in the conditions, but also in the answers. Let's take, for example, the once popular (and still not forgotten) tower of hanoi. This was the name of the puzzle game invented in the 19th century by the French mathematician E. Luc. It contains three rods, one of which is attached n disks with a hole in the middle of each. The diameters of all the 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.

You need to move this tower to another rod, observing the following rules: transfer the disks strictly one at a time (removing the top disk from any rod) and always place 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 proven by induction.

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

Here's another interesting problem:

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

Let's check the smallest numbers first. It is clear that the number 1 in this form cannot be represented. But all odd numbers that are greater than 1 can, of course, be imagined. In fact, any odd 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 this is true for all even numbers? Alas, the next even number refutes our assumption: 6 = 1 + 2 + 3. But the number 8 again does not lend itself. True, the following 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 submitted in the specified form only powers of two. Is this true for the rest of the numbers? It turns out yes! In fact, consider the sum of all natural numbers from m before n inclusive. Since, according to the condition, there are at least two of them, then n > m. As is known, the sum of successive terms arithmetic progression(and this is exactly 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 is inclusively divisible by an odd number greater than 1 and therefore cannot be a power of two. So now it’s clear why it was not possible to represent powers of two in the required form.

It remains to make sure that not powers of two you can imagine. As for odd numbers, we have already dealt with them above. Let's take any even number that is not a power of two. Let the greatest power of two by which it is divisible be 2 a (a- natural). Then if the number is divided by 2 a, it will work out already odd a number greater than 1, which we write in a familiar form - as 2 k+ 1 (k- also natural). This means that in general our even number that 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, average of which is equal to 2 a. It's easy to see that then least of which equals 2 a–k, and the greatest is 2 a + k, and the smallest (and, therefore, all the rest) is positive, i.e. truly 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 average number, because the number of numbers is even, but indicate a couple of medium ones numbers is possible: let these be numbers k And k+ 1. Then least of all numbers equal k+ 1 – 2a(and also positive!), and the greatest is equal to k+ 2a. Their sum is also 2 a(2k + 1).

That's all. So, the answer is: unrepresentable numbers are powers of two, and only those.

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 continuous fence made of N boards. According to Aunt Polly's order, Tom Sawyer whitewashes the fence, but according to his own system: moving clockwise all the time, he first whitewashes an arbitrary board, then skips one board and whitewashes the next one, then skips two boards and whitewashes the next one, then skips three boards and whitewashes the next one, and so on, each time skipping one more board (in this case, some boards can be whitewashed several times - this does not bother Tom).

Tom believes that with such a scheme, sooner or later all the boards will be whitewashed, and Aunt Polly is sure that at least one board will remain unwhitened, no matter how much Tom works. For what N is Tom right, and for what N is Aunt Polly right?

The described whitewashing system seems quite chaotic, so it may initially seem that for anyone (or almost any) N Each board will someday get its share of lime, i.e. mostly, Tom is right. But the first impression is deceiving, 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 unwhitened. The proof of this fact is quite cumbersome (although, in principle, not difficult). We invite the reader to do it himself.

That's what they are - powers of two. On the surface, it’s as simple as shelling pears, but once you dig into it... And we haven’t touched on all the amazing and mysterious properties of this sequence here, but only those that caught our eye. Well, the reader is given the right to independently continue research in this area. They will undoubtedly prove fruitful.

Their number is zero).
And not just twos, as noted earlier!
Those thirsty for details can read the article by V. Boltyansky “Do powers of two often begin with one?” (“Quantum” No. 5, 1978), as well as the article by V. Arnold “Statistics of the first digits of powers of two and the redistribution of the world” (“Quantum” No. 1, 1998).
See problem M1599 from the “Kvant Problem Book” (“Kvant” No. 6, 1997).
There are currently 43 known perfect numbers, the largest of which is 2 30402456 (2 30402457 – 1). It contains over 18 millions numbers

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

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

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

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

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

Let's go back to the example:

And again the formula:

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

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

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

Any number to the zero power is equal to one:

As always, let us ask ourselves: why is this so?

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

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

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

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

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

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

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

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

So, let's formulate a rule:

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

Let's summarize:

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

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

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

Tasks for independent solution:

Well, as usual, examples for independent solutions:

Analysis of problems for independent solution:

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

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

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

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

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

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

Now let's remember the rule about "degree to degree":

What number must be raised to a power to get?

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

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

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

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

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

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

None!

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

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

What about the expression?

But here a problem arises.

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

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

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

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

So if:

• - natural number;
• - integer;

Examples:

Degrees with rational indicator very useful for converting expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

1. Don't forget about the usual properties of degrees:

2. . Here we remember that we forgot to learn the table of degrees:

after all - this is or. The solution is found automatically: .

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- these 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, degree with natural indicator- this is a number multiplied by itself several times;

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

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

By the way, in science a degree with a complex indicator is often used, that is, an indicator is not even real number.

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

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

For example:

Decide for yourself:

Analysis of solutions:

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

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

In this case,

It turns out that:

Answer: .

2. We reduce fractions in exponents to the same form: either both decimals or both ordinary ones. We get, for example:

Answer: 16

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

ADVANCED LEVEL

Determination of degree

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

• — degree base;
• - exponent.

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

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

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

If the exponent is positive integer number:

Construction to the zero degree:

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

If the exponent is negative integer number:

(because you can’t divide by).

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

Examples:

Power with rational exponent

• - natural number;
• - integer;

Examples:

Properties of degrees

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

Let's see: what is and?

A-priory:

So, on the right side of this expression we get the following product:

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

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

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

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

Under no circumstances can you write that.

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

Let's regroup this work like this:

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

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

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

Power with a negative base.

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

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

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

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

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

And so on ad infinitum: with each subsequent multiplication the sign will change. The following simple rules can be formulated:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. 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 we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

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

We get:

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

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

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

Let's go back to the example:

And again the formula:

So now the last rule:

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

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

Example:

Degree with irrational exponent

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

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

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

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

So what do we do if we see irrational indicator degrees? 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

degree whose exponent is infinite decimal 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!

When the number multiplies itself to myself, work called degree.

So 2.2 = 4, square or second power of 2
2.2.2 = 8, cube or third power.
2.2.2.2 = 16, fourth degree.

Also, 10.10 = 100, the second power of 10.
10.10.10 = 1000, third power.
10.10.10.10 = 10000 fourth power.

And a.a = aa, second power of a
a.a.a = aaa, third power of a
a.a.a.a = aaaa, fourth power of a

The original number is called root powers of this number because it is the number from which the powers were created.

However, it is not entirely convenient, especially in the case high degrees, write down all the factors that make up the degrees. Therefore, a shorthand notation method is used. The root of the degree is written only once, and on the right and a little higher near it, but in a slightly smaller font, it is written how many times the root acts as a factor. This number or letter is called exponent or degree numbers. So, a 2 is equal to a.a or aa, because the root a must be multiplied by itself twice to get the power aa. Also, a 3 means aaa, that is, here a is repeated three times as a multiplier.

The exponent of the first degree is 1, but it is not usually written down. So, a 1 is written as a.

You should not confuse degrees with coefficients. The coefficient shows how often the value is taken as Part the whole. The power shows how often a quantity is taken as factor in the work.
So, 4a = a + a + a + a. But a 4 = a.a.a.a

The power notation scheme has the peculiar advantage of allowing us to express unknown degree. For this purpose, the exponent is written instead of a number letter. In the process of solving a problem, we can obtain a quantity that we know is some degree of another magnitude. But so far we do not know whether it is a square, a cube or another, higher degree. So, in the expression a x, the exponent means that this expression has some degree, although unspecified what degree. So, b m and d n are raised to the powers of m and n. When the exponent is found, number is substituted instead of a letter. So, if m=3, then b m = b 3 ; but if m = 5, then b m =b 5.

The method of writing values ​​using powers is also a big advantage when using expressions. Thus, (a + b + d) 3 is (a + b + d).(a + b + d).(a + b + d), that is, the cube of the trinomial (a + b + d). But if we write this expression after raising it to a cube, it will look like
a 3 + 3a 2 b + 3a 2 d + 3ab 2 + 6abd + 3ad 2 + b 3 + d 3 .

If we take a series of powers whose exponents increase or decrease by 1, we find that the product increases by common multiplier or decreases by common divisor , and this factor or divisor is the original number that is raised to a power.

So, in the series aaaaa, aaaa, aaa, aa, a;
or a 5, a 4, a 3, a 2, a 1;
the indicators, if counted from right to left, are 1, 2, 3, 4, 5; and the difference between their values ​​is 1. If we start on right multiply by a, we will successfully get multiple values.

So a.a = a 2 , second term. And a 3 .a = a 4
a 2 .a = a 3 , third term. a 4 .a = a 5 .

If we start left divide to a,
we get a 5:a = a 4 and a 3:a = a 2 .
a 4:a = a 3 a 2:a = a 1

But this division process can be continued further, and we get a new set of values.

So, a:a = a/a = 1. (1/a):a = 1/aa
1:a = 1/a (1/aa):a = 1/aaa.

The complete row would be: aaaaa, aaaa, aaa, aa, a, 1, 1/a, 1/aa, 1/aaa.

Or a 5, a 4, a 3, a 2, a, 1, 1/a, 1/a 2, 1/a 3.

Here are the values on right from one there is reverse values ​​to the left of one. Therefore these degrees can be called inverse powers a. We can also say that the powers on the left are the inverses of the powers on the right.

So, 1:(1/a) = 1.(a/1) = a. And 1:(1/a 3) = a 3.

The same recording plan can be applied to polynomials. So, for a + b, we get the set,
(a + b) 3 , (a + b) 2 , (a + b), 1, 1/(a + b), 1/(a + b) 2 , 1/(a + b) 3 .

For convenience, another form of writing reciprocal powers is used.

According to this form, 1/a or 1/a 1 = a -1 . And 1/aaa or 1/a 3 = a -3 .
1/aa or 1/a 2 = a -2 . 1/aaaa or 1/a 4 = a -4 .

And in order to make a complete series with 1 as a total difference with exponents, a/a or 1 is considered as something that does not have a degree and is written as a 0 .

Then, taking into account the direct and inverse powers
instead of aaaa, aaa, aa, a, a/a, 1/a, 1/aa, 1/aaa, 1/aaaa
you can write a 4, a 3, a 2, a 1, a 0, a -1, a -2, a -3, a -4.
Or a +4, a +3, a +2, a +1, a 0, a -1, a -2, a -3, a -4.

And a series of only individual degrees will look like:
+4,+3,+2,+1,0,-1,-2,-3,-4.

The root of a degree can be expressed by more than one letter.

Thus, aa.aa or (aa) 2 is the second power of aa.
And aa.aa.aa or (aa) 3 is the third power of aa.

All powers of the number 1 are the same: 1.1 or 1.1.1. will be equal to 1.

Exponentiation is finding the value of any number by multiplying that number by itself. Rule for exponentiation:

Multiply the quantity by itself as many times as indicated in the power of the number.

This rule is common to all examples that may arise during the process of exponentiation. But it is right to give an explanation of how it applies to particular cases.

If only one term is raised to a power, then it is multiplied by itself as many times as indicated by the exponent.

The fourth power of a is a 4 or aaaa. (Art. 195.)
The sixth power of y is y 6 or yyyyyy.
The Nth power of x is x n or xxx..... n times repeated.

If it is necessary to raise an expression of several terms to a power, the principle that the power of the product of several factors is equal to the product of these factors raised to a power.

So (ay) 2 =a 2 y 2 ; (ay) 2 = ay.ay.
But ay.ay = ayay = aayy = a 2 y 2 .
So, (bmx) 3 = bmx.bmx.bmx = bbbmmmxxx = b 3 m 3 x 3 .

Therefore, in finding the power of a product, we can either operate with the entire product at once, or we can operate with each factor separately, and then multiply their values ​​with the powers.

Example 1. The fourth power of dhy is (dhy) 4, or d 4 h 4 y 4.

Example 2. The third power is 4b, there is (4b) 3, or 4 3 b 3, or 64b 3.

Example 3. The Nth power of 6ad is (6ad) n or 6 n a n d n.

Example 4. The third power of 3m.2y is (3m.2y) 3, or 27m 3 .8y 3.

The degree of a binomial, consisting of terms connected by + and -, is calculated by multiplying its terms. Yes,

(a + b) 1 = a + b, first degree.
(a + b) 1 = a 2 + 2ab + b 2, second power (a + b).
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3, third power.
(a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4, fourth power.

The square of a - b is a 2 - 2ab + b 2.

The square of a + b + h is a 2 + 2ab + 2ah + b 2 + 2bh + h 2

Exercise 1. Find the cube a + 2d + 3

Exercise 2. Find the fourth power of b + 2.

Exercise 3. Find the fifth power of x + 1.

Exercise 4. Find the sixth power 1 - b.

Sum squares amounts And differences binomials occur so often in algebra that it is necessary to know them very well.

If we multiply a + h by itself or a - h by itself,
we get: (a + h)(a + h) = a 2 + 2ah + h 2 also, (a - h)(a - h) = a 2 - 2ah + h 2 .

This shows that in each case, the first and last terms are the squares of a and h, and the middle term is twice the product of a and h. From here, the square of the sum and difference of binomials can be found using the following rule.

The square of a binomial, both terms of which are positive, equal to square the first term + twice the product of both terms, + the square of the last term.

Square differences binomials is equal to the square of the first term minus twice the product of both terms plus the square of the second term.

Example 1. Square 2a + b, there is 4a 2 + 4ab + b 2.

Example 2. Square ab + cd, there is a 2 b 2 + 2abcd + c 2 d 2.

Example 3. Square 3d - h, there is 9d 2 + 6dh + h 2.

Example 4. The square a - 1 is a 2 - 2a + 1.

For a method for finding higher powers of binomials, see the following sections.

In many cases it is effective to write down degrees without multiplication.

So, the square of a + b is (a + b) 2.
The Nth power of bc + 8 + x is (bc + 8 + x) n

In such cases, the parentheses cover All members under degree.

But if the root of the degree consists of several multipliers, the parentheses may cover the entire expression, or may be applied separately to the factors depending on convenience.

Thus, the square (a + b)(c + d) is either [(a + b).(c + d)] 2 or (a + b) 2 .(c + d) 2.

For the first of these expressions, the result is the square of the product of two factors, and for the second, the result is the product of their squares. But they are equal to each other.

Cube a.(b + d), is 3, or a 3.(b + d) 3.

The sign in front of the members involved must also be taken into account. It is very important to remember that when the root of a degree is positive, all its positive powers are also positive. But when the root is negative, the values ​​with odd powers are negative, while the values even degrees are positive.

The second degree (- a) is +a 2
The third degree (-a) is -a 3
The fourth power (-a) is +a 4
The fifth power (-a) is -a 5

Hence any odd the degree has the same sign as the number. But even the degree is positive regardless of whether the number has a negative or positive sign.
So, +a.+a = +a 2
And -a.-a = +a 2

A quantity that has already been raised to a power is raised to a power again by multiplying the exponents.

The third power of a 2 is a 2.3 = a 6.

For a 2 = aa; cube aa is aa.aa.aa = aaaaaa = a 6 ; which is the sixth power of a, but the third power of a 2.

The fourth power of a 3 b 2 is a 3.4 b 2.4 = a 12 b 8

The third power of 4a 2 x is 64a 6 x 3.

The fifth power of (a + b) 2 is (a + b) 10.

The Nth power of a 3 is a 3n

The Nth power of (x - y) m is (x - y) mn

(a 3 .b 3) 2 = a 6 .b 6

(a 3 b 2 h 4) 3 = a 9 b 6 h 12

The rule applies equally to negative degrees.

Example 1. The third power of a -2 is a -3.3 =a -6.

For a -2 = 1/aa, and the third power of this
(1/aa).(1/aa).(1/aa) = 1/aaaaaa = 1/a 6 = a -6

The fourth power of a 2 b -3 is a 8 b -12 or a 8 /b 12 .

The square is b 3 x -1, there is b 6 x -2.

The Nth power of ax -m is x -mn or 1/x.

However, we must remember here that if the sign previous degree is "-", then it must be changed to "+" whenever the degree is an even number.

Example 1. The square -a 3 is +a 6. The square of -a 3 is -a 3 .-a 3, which, according to the rules of signs in multiplication, is +a 6.

2. But the cube -a 3 is -a 9. For -a 3 .-a 3 .-a 3 = -a 9 .

3. The Nth power -a 3 is a 3n.

Here the result can be positive or negative depending on whether n is even or odd.

If fraction is raised to a power, then the numerator and denominator are raised to a power.

The square of a/b is a 2 /b 2 . According to the rule multiplying fractions,
(a/b)(a/b) = aa/bb = a 2 b 2

The second, third and nth powers of 1/a are 1/a 2, 1/a 3 and 1/a n.

Examples binomials, in which one of the terms is a fraction.

1. Find the square of x + 1/2 and x - 1/2.
(x + 1/2) 2 = x 2 + 2.x.(1/2) + 1/2 2 = x 2 + x + 1/4
(x - 1/2) 2 = x 2 - 2.x.(1/2) + 1/2 2 = x 2 - x + 1/4

2. The square of a + 2/3 is a 2 + 4a/3 + 4/9.

3. Square x + b/2 = x 2 + bx + b 2 /4.

4 The square of x - b/m is x 2 - 2bx/m + b 2 /m 2 .

It was previously shown that fractional coefficient can be moved from the numerator to the denominator or from the denominator to the numerator. Using the scheme for writing reciprocal powers, it is clear that any multiplier can also be moved, if the sign of the degree is changed.

So, in the fraction ax -2 /y, we can move x from the numerator to the denominator.
Then ax -2 /y = (a/y).x -2 = (a/y).(1/x 2 = a/yx 2 .

In the fraction a/by 3, we can move y from the denominator to the numerator.
Then a/by 2 = (a/b).(1/y 3) = (a/b).y -3 = ay -3 /b.

In the same way we can move a factor that has a positive exponent into a numerator or factor with negative degree into the denominator.

So, ax 3 /b = a/bx -3. For x 3 the inverse is x -3 , which is x 3 = 1/x -3 .

Therefore, the denominator of any fraction can be removed entirely, or the numerator can be reduced to one, without changing the meaning of the expression.

So, a/b = 1/ba -1 , or ab -1 .