Indices

SURDS AND INDICES

IMPORTANT FORMULAE

1. LAWS OF INDICES:

(i) a^m x aⁿ = a^{m + n}

(ii) a^m_­ / aⁿ = a^{m - n}

(iii) ( a^m )ⁿ = a^mn

(iv) ( a b )ⁿ = aⁿbⁿ



(v) ( a / b )ⁿ = ( aⁿ / bⁿ )

(vi) a⁰ = 1



2. SURDS: Let a be a rational number and n be a positive integer such that a^1/n = ⁿsqrt(a)

is irrational. Then ⁿsqrt(a) is called a surd of order n.

3. LAWS OF SURDS:

(i) ⁿ√a = a^1/2

(ii) ⁿ √ab = ⁿ √a * ⁿ √b

(iii) ⁿ √a/b = ⁿ √a / ⁿ √b

(iv) (ⁿ √a)ⁿ = a

(v) ^m√(ⁿ√(a)) = ^mn√(a)

(vi) (ⁿ√a)^m = ⁿ√a^m

Examples:

Ex. 1. Simplify : (i) (27)^2/3 (ii) (1024)^-4/5 (iii)( 8 / 125 )^-4/3

Sol . (i) (27)^2/3 = (3³)^2/3 = 3^{( 3 * ( 2/ 3))} = 3² = 9

(ii) (1024)^-4/5 = (4⁵)^-4/5 = 4 ^{{ 5 * ( (-4) / 5 )}} = 4^-4 = 1 / 4⁴ = 1 / 256

(iii) ( 8 / 125 )^-4/3 = {(2/5)³}^-4/3 = (2/5)^{{ 3 * ( -4/3)}} = ( 2 / 5 )^-4 = ( 5 / 2 )⁴ = 5⁴ / 2⁴

Ex. 2. Evaluate: (i) (.00032)^3/5 (ii)l (256)^0.16 x (16)^0.18.

Sol. (i) (0.00032)^3/5 = ( 32 / 100000 )^3/5. = (2⁵ / 10⁵)^3/5 = {( 2 / 10 )⁵}^3/5 = ( 1 / 5 )^{(5 * 3 / 5)} = (1/5)³ = 1 / 125

(ii) (256)^{0. 16} * (16)^{0. 18} = {(16)²}^{0. 16} * (16)^{0. 18} = (16)^{(2 * 0. 16)} * (16)^{0. 18}

=(16)^0.32 * (16)^0.18 = (16)^(0.32+0.18) = (16)^0.5 = (16)^1/2 = 4.

Ex. 3. Find the largest from among 4√6, √2 and ³√4.

Sol. Given surds are of order 4, 2 and 3 respectively. Their L.C,M, is 12, Changing each to a surd of order 12, we get:

⁴√6 = 6^1/4 = 6^{((1/4)*(3/3))} = 6^3/12 = (6³)^1/12 = (216)^1/12.

√2 = 2^1/2 = 2^{((1/2)*(6/6))} = 2^6/12 = (2⁶)^1/12 = (64)^1/12.

³√4 = 4^1/3 = 4^{((1/3)*(4/4))} = 4^4/12 = (4⁴)^1/12 = (256)^1/12.

Clearly, (256)^1/12 > (216)^1/12 > (64)^1/12

Largest one is (256)^1/12. i.e. ³√4 .