Square root from non-negative number a is called such non-negative number b that b^2 = a. Extraction of a square root — more difficult task, than squaring, but for its decision exists a set of methods.
Instruction
1. If b is a square root from a, then, generally speaking, (-b) too can be considered as that as (-b) ^2 = b^2. However in practice a square root it is considered to be only non-negative number.
2. For a rough estimate of size of a square root it is possible to use the table of squares. Having defined between what values of squares there is a set number, thereby define borders between which there is a size of a square root. For example, number 138 is less, than 144 = 12^2, but it is more, than 121 = 11^2. Therefore, the square root from it has to lie between numbers 11 and 12. The approximate value 11.7 when squaring yields result 136.89, and approximate value 11.8 — number 139.24.
3. If there is no table of squares near at hand, or the set number goes beyond its limits, it is possible to use the theorem saying that the sum of odd numbers from 1 to 2n+1 is always a full square of number n + 1. Really, 1^2 = 1, and for any n always n^2 + 2n + 1 = (n + 1) ^2 on the known formula of a square of the sum. Thus if consistently to subtract all odd numbers from the set number, since unit until the result of subtraction becomes zero or will be taken less next deductible, then the quantity of steps of this procedure will be equally whole part of a square root. If further specification is required, then it can be made simple selection, as in the previous option.
4. Absolutely rough estimate of size of a square root from very large number is in certain cases necessary. Such assessment can be constructed, proceeding from the number of figures in the set number. If this quantity nechetno, that is to equally some 2n, then the root is approximately equal 6*10^n. If the number of figures chetno, then it is possible to take number 2*10^n for a rough estimate.
5. It is possible to apply the iterative method known to more exact calculation of a square root as Heron's formula. Let it is required to take a root from number a. Let's take initial x0 = to a. Further steps are calculated on a formula: x (n+1) = (xn + a/xn)/2. If n → ∞, then xn → √a. As at calculations on this formula x1 = (a + 1)/2, it makes sense to begin with this value at once.