Radicals are an irrational form that equate to a rational or irrational number or expression. They can be expressed as fractional exponents that provides the convenience of not requiring a radical symbol. Most math of radicals is to change them to their simplest form to reduce complexity.

n√ a is the general notation for the nth root of “a”. If n = 2 the index is not usually written, a square root is assumed. “a” is the radicand. The √ symbol denotes a radical.

How Radicals equate to Exponents.

The principal nth root of a positive number is the positive root. The square root of 9 has 2 root numbers 3 and -3, only 3 is a principal root.

The principal nth root of zero is zero. Just says any root of zero is zero.

The principal nth root of a negative number is the negative root when n is odd. The cube root of -27 is -3, because the index 3 is odd:

3√ -27 = -3, because the index is an odd number: (-3) (-3) (-3) = -27

√ -25 = nonsense. A square root is an even number index. Whether (5) (5) is multiplied or (-5) (-5), the result is positive 25.

-√ 25 = -5. The square root of 25 is 5 or -5, the minus sign outside the radical indicates the negative of the root.

2√ 16 = 4

3√ -8 = -2

n√ 0 = 0

Radicals can be expressed as fractional exponents, and fractional exponents can be transformed to irrational radical form. The index of the radical, “n”, becomes the denominator of the fractional exponent:

n√ a = a1/n

n√ am = am/n

Examples:

√ x2 = x2/2 = x1 = x

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(125m)2/3 = 25 3√ m2

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3√ (ab)2 = (ab)2/3

3√ 272 = (3)2 = 9

3√ 272 = 272/3 = 9

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3√ y27 = y27/3 = y9

Let y = 3, the cube root of 327 = 19,683 = 39

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3√ x6 = x6/3 = x2

Let x =2, the cube root of 26 = 4 = 22

3√ (-27 x6 y3) = -271/3 x6/3 y3/3 = -3 x2 y, because (-3 x2 y)3 = -27 x6 y3

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