Square root from number x call number a which at multiplication by itself gives number x: a * a = a^2 = x, √x = a. As well as over any numbers, over square roots it is possible to carry out arithmetic operations of addition and subtraction.
Instruction
1. First, at addition of square roots try to take these roots. It will be possible if numbers under the sign of a root are full squares. For example, let expression √4 + √9 is set. The first 4 is a square of number 2. The second 9 is a square of number 3. Thus it turns out that: √4 + √9 = 2 + 3 = 5.
2. If under the sign of a root there are no full squares, then try to take out a number multiplier from under the sign of a root. For example, let expression √24 + √54 is given. Factorize numbers: 24 = 2 * 2 * 2 * 3, 54 = 2 * 3 * 3 * 3. In number 24 there is a multiplier 4 which can be taken out from under the sign of a square root. In number 54 - a multiplier 9. Thus, it turns out that: √24 + √54 = √ (4 * 6) + √ (9 * 6) = 2 * √6 + 3 * √6 = 5 * √6. In this example as a result of carrying out of a multiplier from under the sign of a root it turned out to simplify the set expression.
3. Let the sum of two square roots be a fraction denominator, for example, of A / (√a + √b). And let you are faced by a task "to get rid of irrationality in a denominator". Then it is possible to use the following method. Increase numerator and a denominator of fraction by expression of √a - √b. Thus in a denominator formula abridged multiplication will turn out: (√a + √b) * (√a - √b) = a – b. By analogy if in a denominator the difference of roots is given: √a - √b, numerator and a denominator of fraction needs to be increased by expression of √a + √b. For an example, let the fraction 4 / (√3 + √5) = 4 * (√3-√5) / ((√3 + √5) * (√3-√5)) = 4 * (√3-√5) / (-2) = 2 * is given (√5-√3).
4. Review more difficult example of disposal of irrationality in a denominator. Let the fraction 12 / be given (√2 + √3 + √5). It is necessary to increase numerator and a denominator of fraction by expression √2 + √3 - √5:12 / (√2 + √3 + √5) = 12 * (√2 + √3-√5) / ((√2 + √3 + √5) * (√2 + √3-√5)) = 12 * (√2 + √3-√5) / (2 * √6) = √6 * (√2 + √3-√5) = 2 * √3 + 3 * √2-√30.
5. And at last, if you need only approximate value, then it is possible to count values of square roots on the calculator. Calculate values separately for each number and write down with a necessary accuracy (for example, two signs after a comma). And then make the required arithmetic operations, as with usual numbers. For example, let it is necessary to learn approximate value of expression √7 + √5 ≈ 2.65 + 2.24 = 4.89.