We have already understood to ourselves that every mathematical action corresponds to a similar, but opposite in direction action.

For addition, such an inverse action is subtraction, for multiplication - division. Now let's try to figure out which action is the opposite for exponentiation. Since **exponentiation** is a multiple multiplication, then, obviously, the inverse action will be multiple division.

For example, 32 can be divided by 2 and get 16, then 16 divided by 2 and get 8; then 8 divided by 2 and get 4; then 4 divided by 2 and get 2; finally, then 2 is divided by 2 and 1 is obtained. In short, these actions can be written as 32: 2: 2: 2: 2: 2 = 1. (Our __ task __ was to get to 1.) Since we made the division 5 times and got to 1, we can say that 2 is the fifth root of 32.

If we consider the number 81, we see that 81: 3: 3: 3: 3 = 1, so 3 is the fourth-degree root of 81. (Why, actually, the root? Where did this word come from? This can be explained in this way : the number 32 grows from the base 2, and 81 grows from the base 3 just as the plant grows from the roots.)

Such a mathematical operation is denoted as $ \ sqrt {} $. The number of roots is indicated by the number in the upper left part of the root. So, the fifth degree root from 32 can be written as $ \ sqrt [5] {32} $, the fourth power root from 81 can be written as $ \ sqrt [4] {81} $. The $ \ sqrt {} $ icon is called the radical sign, and the numbers containing the roots are called **radicals** . The word "radical" came to us from Latin, where it simply means "root."

We rarely meet with roots of high degrees, most often we have to deal with operations that are inverse to the construction of a second degree, that is, a square. The extraction of the second degree root is called the square root extraction, and $ \ sqrt [2] {} $ is called the **square root** , and the two on the left is often omitted. In the future, under the $ \ sqrt {} $ icon without the number in the upper left corner, we will always mean the square root.

What is the square root of a number? 25 is square 5, so we can say that 5 is the square root of 25, or $ \ sqrt {25} = 5 $. Therefore, one should say “five is the second-degree root of 25,” but the term “square root” is usually used. (Similarly, a third degree root is called a **cubic root** .)

The next problem is to figure out how to find the root of such and such a certain number. Here you can go from the opposite. Suppose we know that $ 2 ^ 5 = $ 32, which means that if 32 is divisible by 2 times by 2, the result is 1. (If we raise a number to some degree, it is not difficult to go in the opposite order.)

In practice, the arithmetic **method for determining the roots** is a series of inverse actions. Let's try to extract the square root of 625. The calculation scheme will be as follows:

The first digit of the answer, 2, we get a selection. We know that 2 × 2 = 4, this is the nearest possible number, less than 6, because 3 × 3 = 9, which is more than 6. Then we subtract and put two digits instead of one, as is customary with the usual division into a bar. (If we extracted the cube root, we would endure three digits, in the case of a fourth root root, four digits, and so on.) To get the next digit, you must divide 225 by 45. You get 45, doubling the first digit of the answer, which gives you 4. The second digit must be equal to the second digit of your answer, so you can also find it by fitting, so that you get the number closest to 225. The number 5 fits most accurately, since 5x45 = 225.

This process may seem very difficult to you, and you will be absolutely right. It is very difficult to calculate the **number roots by the** arithmetic method, but the results turn out to be useful for various calculations.

Consider the following example. What is $ \ sqrt {2} $? What number should be **squared** to get 2?

We can immediately determine that among the integers there is no such number, because 1 × 1 = 1, and 2x2 = 4. The first number is too small, and the second is too large. Therefore, the answer will be a fractional number.

And can there even exist a square root in the form __ fractional numbers __ ? Why not? According to our definition of exponential expressions, $ (1 \ frac25) ^ 2 $ is $ 1 \ frac25 \ times 1 \ frac25 $, and the answer is the number $ 1 \ frac {24} {25} $. And this, in turn, means that $ \ sqrt {1 \ frac {24} {25}} $ is $ 1 \ frac {2} {5} $. Now we are convinced that not only the square root can be a fractional number. And in both cases the same rules are true as in the case of integers.

In addition, it turned out by chance that the number $ 1 \ frac {2} {5} $, being multiplied by itself, gives a result close to 2. It follows that $ 1 \ frac {2} {5} $ is close to $ \ sqrt {2} $. Only $ \ frac {1} {25} $ separates us from the desired answer, since $ (1 \ frac {2} {5}) ^ 2 $ is $ 1 \ frac {24} {25} $, and we need get the number $ 1 \ frac {25} {25} $, that is, 2.

But you can get a more accurate answer. If we multiply the fractional number $ 1 \ frac {41} {100} $ by ourselves, we get $ 1 \ frac {9881} {10000} $, which is much closer to 2. It may seem that if we make more accurate calculations, we are early or later we will find the exact value of the fractional number, which is the **square root** of 2, although it may be a very complex number.

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Why, actually, the root?Where did this word come from?

What is the square root of a number?

What is $ \ sqrt {2} $?

Why not?