Question

Find the value of n, where n is an integer and 2^n–5 × 6^2n–4 = `1/(12^4 xx 2)`.

Answer 1

We have, 2^n–5 × 6^2n–4 = `1/(12^4 xx 2)`

⇒ `2^n/2^5 xx 6^(2n)/6^4 = 1/(12^4 xx 2)`  ......`[∵ a^(m-n) = a^m/a^n]`

⇒ `(2^n xx 6^(2n))/(2^5 xx 6^4) = 1/((2 xx 6)^4 xx 2)`  ......[∵ 12 = 6 × 2]

⇒ 2ⁿ × (6²)ⁿ = `(2^5 xx 6^4)/(2^4 xx 6^4 xx 2)`  ......[By cross-multiplication] [∵ a^mn = (a^m)ⁿ and (a × b)^m = a^m × a^m+n]

⇒ 2ⁿ × 36ⁿ = `(2^5 xx 6^4)/(2^5 xx 6^4)`  ......[∵ a^m × aⁿ = a^m+n]

⇒ 2ⁿ × 36ⁿ = 1

⇒ (2 × 36)ⁿ = 1  ......[∵ a^m × b^m = (ab)^m]

⇒ (72)ⁿ = (72)⁰  ......[∵ a⁰ = 1]

∴ n = 0  ......[∵ If a^m = aⁿ ⇒ m = n]