Radicals and Rational Indices

If x 2 = a, then x is a square root of a. Radicals Do you remember what square root means? For example: 3 2 = 9 3 is a square root of = 49 7 is a square root of 49. (  3) 2 = 9  3 is a square root of 9. Note that both 3 and  3 are square roots of 9.

How about x 3 = a? If x 3 = ax is a cube root of a. If x 4 = a x is a fourth root of a. If x 5 = a x is a fifth root of a. 5 3 = 1255 is a cube root of = (  2) 4 = 162 and  2 are fourth roots of = is a fifth root of For example:

If n is a positive integer such that x n = a, then x is an n th root of a. denote n th root of a For example: It should be noticed that:  positive fourth root of 16  negative fourth root of 16 Radical

Find the value of. Follow-up question

We learnt that (a m ) n = a mn for integral indices. Suppose the law also hold for rational indices, then Rational Indices How about the index is a rational number, for example ?

By definition, is the square root of a., where n is a positive integer. In general, By definition, is the cube root of a.

where m and n are integers and a > 0, n >0. To summarize, we have Then, what is the meaning of ? and For example,

Find the value of.

Follow-up question

Simplify and express your answer with positive index.

Simplify and express your answer with positive index. Follow-up question