Laws of Indices –
For all real numbers a and b and all rational numbers m and n, we have –
i) aᵐ X aⁿ = aᵐ⁺ⁿ
aᵐ
ii) ------ = aᵐˉⁿ, a ≠ 0
aⁿ
iii) (aᵐ)ⁿ = aᵐⁿ
iv) (ab)ᵐ = aᵐ . bᵐ
a aⁿ
v) (------)ⁿ = ------, b ≠ 0
b bⁿ
1
vi) aˉᵐ = ------- , a ≠ 0
aᵐ
a 1 a b b
vii) (-----)ˉⁿ = --------- = 1 ÷ (-----)ⁿ = 1 X (-----)ⁿ = (-----)ⁿ
b a b a a
(-----)ⁿ
b
viii) aⁿ = aᵐ, if m = n, where a > 0, a ≠ 1
ix) aᵐ = bᵐ, where m ≠ 0, a = b, and a & b are positive
x) pᵐ X qⁿ X rᵒ = pᵃ qᵇ rᶜ, when m = a, n = b, and o = c, where p, q, r are different primes.