Laws of Rational Indices

Do you remember what a square root means? If x2 = a, then x is a square root of a. Example: 42 = 16 4 is a square root of 16. (4)2 = 16 4 is a square root of 16. Both 4 and 4 are the square roots of 16.

How about if the index is not equal to 2? If x3 = a x is the cube root of a. If x4 = a x is a fourth root of a. If x5 = a x is the fifth root of a. ● ● ● If xn = a x is an nth root of a.

if xn = a, then x is an n th root of a. In general, for any positive integer n, if xn = a, then x is an n th root of a. Radical n a denote n th root of a ‘ ’ is the radical sign. n

a In general, for any positive integer n, if xn = a, then x is an n th root of a. Radical n a denote n th root of a Example: 43 = 64 64 = 4 3 The cube root of 64 is 4. 24 = 16 16 = 2 4 A fourth root of 16 is 2. 16 = 2 4  (2)4 = 16 Another fourth root of 16 is 2.

a In general, for any positive integer n, if xn = a, then x is an n th root of a. Radical n a denote n th root of a The ‘’ sign denotes the negative 4th root. Example: 43 = 64 64 = 4 3 The cube root of 64 is 4. 24 = 16 16 = 2 4 A fourth root of 16 is 2. 16 = 2 4  (2)4 = 16 Another fourth root of 16 is 2.

a a In fact, we have the following 4 cases for xn = a. (i) When n is odd and a > 0: a has only one positive nth root . n a e.g. 125 = 5, 3 53 = 125 which is positive. (ii) When n is odd and a < 0: a has only one negative nth root . n a 3 e.g. 125 = 5, (5)3 = 125 which is negative.

a a In fact, we have the following 4 cases for xn = a. (iii) When n is even and a > 0: a has two nth roots and  . n a e.g. 16 = 2, which is positive. 4 4  16 = 2, which is negative. 24 = (2)4 = 16 (iv) When n is even and a < 0: is not a real number. n a e.g. 100 , 16 are not real numbers. 4

We can also use a calculator to find the values of radicals. Follow-up question Find the value of . 3 216 We can also use a calculator to find the values of radicals. 216 6 =  ∵ 216 6 3 = 6 216 3 = ∴

Follow-up question 216 Find the value of . By keying in: 6 216 = ∴ 3 216 By keying in: gives 6 216 3 = ∴

In junior forms, we have learnt the definitions and laws of integral indices.

Zero and negative integral indices If a  0, then (i) a0 = 1 (ii) an = 1 an n is a positive integer. Laws of integral indices If m and n are integers and a, b  0, then (ii) am  an = = am  n am an (i) am  an = am + n (iii) (am)n = am  n (iv) (ab)n = anbn (v) n b a = ø ö ç è æ

How about if the index is a rational number, for example ? 3 2 1 , - a The meaning of Using the law: (am)n = am  n, we have ∵ 2 1 ) ( a  = a = 2 1 a = ∴ is the square root of a. 2 1 a

a > 0 and n is a positive integer. ∵ 3 1 ) ( a  = 3 1 a = a = ∴ is the cube root of a. 3 1 a By similar argument, ∵ ∴ is the nth root of a. a > 0 and n is a positive integer.

where a > 0, m is an integer and n is a positive integer. The meaning of By the laws of indices and the above result, and Hence, we have where a > 0, m is an integer and n is a positive integer. For example,

Zero and negative rational indices Now, we can extend the definitions and laws of indices to rational indices p and q. Zero and negative rational indices If a > 0, then p is a rational number. (i) a0 = 1 (ii) ap = 1 ap

Laws of rational indices For rational numbers p and q, and positive numbers a and b: (ii) ap  aq = = ap  q ap aq (i) ap  aq = ap + q (iii) (ap)q = ap  q (iv) (ab)p = apbp (v) p b a = ø ö ç è æ

Try to find the value of . Alternative Solution

value of with a calculator. We can also find the value of with a calculator. By keying in: gives ∴

Follow-up question Find the value of . Alternative Solution

Follow-up question Find the value of . Using calculator, by keying in: gives ∴