The arithmetic root n-y of degree from real number a call such non-negative number x, n-I degree of which am equal to number a. I.e. (√n) a = x, x^n = a. There are various ways of addition of an arithmetic root and a rational number. Here square roots will be for descriptive reasons considered (or square roots), explanations will be complemented with examples with calculation of roots of other degrees.
Instruction
1. Let expressions of a type of a + √b be set. The first that needs to be made, is to define whether number b is a full square. I.e. to try to find such number c that c^2 = b. In this case you take a square root from number b, receive number c and put it with number a: a + √b = a + √ (c^2) = a + page. If you deal not with a square root, and with the degree root n-y, then from under the sign of a root it is necessary for full extraction of number b that this number was n-y degree of some number. For example, number 81 will be taken from under a square root: √81 = 9. Also it will be taken from under the sign of a fourth root: (√4) 81 = 3.
2. Pay attention to the following examples.· 7 + √25 = 7 + √ (5^2) = 7 + 5 = 12. Here under the sign of a square root there is number 25 which is a full square of number 5.• 7 + (√3) 27 = 7 + (√3) (3^3) = 7 + 3 = 10. Here the cubic root was taken from number 27 which is a cube of number 3.• 7 + √ (4/9) = 7 + √ ((2/3) ^2) = 7 + 2/3 = 23/3. For extraction of a root from fraction it is necessary to take a root from numerator and from a denominator.
3. If number b under the sign of a root is not a full square, then try to factorize it and to take out the multiplier which is a full square from under the sign of a root. I.e. let number b has b appearance = c^2 * d. Then √b = √ (c^2 * d) = with * √d. Or number b may contain squares of two numbers, i.e. b = c^2 * d^2 * e * f. Then √b = √ (c^2 * d^2 * e * f) = with * d * √ (e * f).
4. Examples of removal of a multiplier from under the sign of a root: • 3 + √18 = 3 + √ (3^2 * 2) = 3 + 3√2 = 3 * (1 + √2). • 3 + √ (7/4) = 3 + √ (7/2^2) = 3 + √7/2 = (6 + √7) / 2. In this example the full square from a denominator of fraction.· 3 + (√4)240 = 3 + (√4) (2^4 * 3 * 5) = 3 + 2 * (√4) 15 was taken out. Here it turned out to take out 2 in the fourth degree from under the sign of a fourth root.
5. And at last, if you need to receive approximate result (in case the radicand is not a full square), use the calculator for calculation of value of a root. For example, 6 + √7 ≈ 6 + 2.6458 = 8.6458.