How to solve the roots?

I really do not like some studentsequations and problems in which the sign of the root occurs. But it is not so difficult to solve the root example, it is important to know which side to approach to the problem. The icon itself, which denotes extraction of the root, is called a radical. How to solve the roots? Extract the square root of the number - this means, pick up a number that in the square will give that same value under the sign of the radical.

So, how to solve the square roots

Solving square roots is not difficult. For example, you need to find out how much the root will be from 16. In order to solve this simple example, you need to remember how much there will be 2 in the box - 2², then 3², and, finally, 4². Only now we see that the result (16)corresponds to the request. That is, in order to extract the root, we had to select the possible values. It turns out that in order to solve the roots, there is no precise and verified algorithm. To facilitate the work of the "solver", mathematicians are recommended to memorize (by heart, as a multiplication table) the values ​​of the squares of numbers to twenty. Then you can easily extract the root from the numbers that are more than a hundred. And, conversely, to see immediately that the root of this number can not be extracted, that is, the answer will not have an integer.

We figured out how to solve square roots. And now let's figure out what square root solutions do not have. For example, negative numbers. Here it is clear that if two negative numbers are multiplied, the answer is obtained with a plus sign. Further that it is necessary to know. The root can be extracted from any number (except negative, as mentioned above). Just the answer can turn into a decimal fraction. That is, contain a certain number of digits after the decimal point. For example, the root of two has the value 1.41421 and it's not all the digits after the decimal point. Such values ​​are rounded to facilitate calculations, sometimes to the second digit after the comma, sometimes to the third or fourth. In addition, it is often practiced to leave the number under the root as an answer if it looks good and compact. It's so clear what it means.

How to solve equations with roots?

In order to solve equations with roots, one needs to apply one of the techniques we devised not by us. For example, raise both sides of such an equation to a square. For example:

The root of X + 3 = 5

Let's square the left and right sides of the equation:

X² + 9 = 25

Now you can see how to solve this equation. First we find out what X is equal to² (and it is equal to 16), and then we extract the root from it. Answer: 4. However, here it is worth saying that this equation actually has two solutions, two roots: 4 and -4. After all, -4 in the square will also give 16.

In addition to this method, sometimes it is more attractive and convenient to replace a variable that is under the root - another variable, in order to get rid of this root.

Y = the root of X.

Subsequently, solving the equation, we return to the replacement and finish the calculations with the root.

That is, we get X = Y². And this will be the solution.

It should be said that there are several methods of solving equations with roots.

How to solve the roots in degree?

The radical, in the basis of which there is no degree,means that you need to extract from the expression or number the square root, that is, the square degree vice versa. It's simple and understandable. For example: the root of 9 = 3, (a 3² = 9), the root of 16 = 4 (4² = 16) and all in the same spirit. But what does it mean if the root has a degree? This means that, again, it is necessary to carry out an action opposite to the erection to this very degree. For example, you need to know the value of the root of the cubic of 27.
For this, it is necessary to choose a number that, when erected in a cube, will give 27. This is 3 (3 * 3 * 3 = 27).

Hence:

root ³ of 27 = 3

Similar actions must be carried out if the degree of the root is 4, 5. Only in this case it is necessary to choose a number which, when raised to the power n will give a value under the root ndegree.

Here it is necessary to say that the degrees of roots and degreeThe subordinate expressions can be reduced. However, according to the rules. If the number or variable under the root has a degree that is a multiple of the degree of the root, they can be shortened. For example:

root³ from X⁶ = X²

These rules of action with roots and degrees are simple, you need to know them clearly, and then the calculation will be simple. How to solve the roots in degree, we figured out, now we move on.

How to solve the root under the root?

This horrible expression of the root under the root at firsta look that can not be solved. But to correctly calculate the value of such an expression, you need to know the properties of the roots. In this case, you just need to replace two roots - one. For this, the degree of these radicals must be simply multiplied. For example:

root³from the root 729 = (root³* root²) of 729

That is, here we multiplied the root of the cube to the square root. As a result, we obtained a sixth root:

root⁶ of 729 = 3

Similarly, it is necessary to solve other similar roots under the root.

Having considered all the proposed examples, it is easyto agree that the solution of the roots is not such a difficult task. Of course, when it comes down to simple, banal arithmetic, sometimes it's easier to use the usual calculator. However, before making calculations, you need to do everything possible to simplify your task by minimizing the number and complexity of arithmetic calculations. Then the solution will become simple and, most importantly, interesting.