How to fold the root and the number

Arithmetic root of n-th degree of a real number a is called non-negative number x, n-I the degree of which is equal to the number a. Ie (√n) a = x, x^n = a. There are various ways of addition and the arithmetical root of a rational number. Here for greater clarity, will be considered the roots of the second degree (or square roots), explanations will be complemented by examples of calculation of roots of other degrees.

Instruction

1. Let the set of expressions of the form a + √b. The first thing to do is to determine whether the number b is a perfect square. Ie try to find such a number c that c^2 = b. In this case, you extract the square root of the number b, get c and put him with number a: a + √b = a + √(c^2) = a + S. If you are not dealing with the square root, and the root of n-th degree, then complete extraction of the number b from under the sign of the root is necessary to this number was the nth degree of a certain number. For example, the number 81 will be removed from under the square root: √81 = 9. Also it will be removed from under the sign of the fourth root: (√4) 81 = 3.
2. Note the following examples.
   
   • 7 + √25 = 7 + √(5^2) = 7 + 5 = 12. Here under the sign of the square root is the number 25, which is the full square of 5.
   
   • 7 + (√3)27 = 7 + (√3) (3^3) = 7 + 3 = 10. Here were extracted the cube root of the number 27 which is the cube of the number 3.
   
   • 7 + √(4/9) = 7 + √( (2/3)^2 ) = 7 + 2/3 = 23/3. To extract the square root of a fraction you must remove the root from the numerator and the denominator.
3. If the b number under the root sign is not a perfect square, then try to decompose it into factors and make the multiplier, which is a complete square, under the sign of the root. Ie let the number of b has the form b = c^2 * d. Then √b = √(c^2 * d) = c * √d. Or b can contain the squares of two numbers, i.e. b = c^2 * d^2 * e * f . Then √b = √(c^2 * d^2 * e * f) = c * d * √(e * f).
4. Examples of sentencing multipliers under the sign of the 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, was handed down the full square of the denominator of a fraction.
   
   • 3 + (√4)240 = 3 + (√4) (2^4 * 3 * 5) = 3 + 2 *(√4) 15. There turned out to take 2 in the fourth degree under the sign of the fourth root.
5. And finally, if you need to obtain the approximate result (if the radical expression is a perfect square), use the calculator to compute the value of the root. For example, 6 + √7 ≈ 6 + 2,6458 = 8,6458.