Question

Find the value of n when:

(i) 5²ⁿ × 5³ = 5⁹
(ii) 8 × 2ⁿ⁺² = 32
(iii) 6²ⁿ⁺¹ ÷ 36 = 6³

Answer 1

(i) 5²ⁿ × 5³ = 5⁹
5²ⁿ⁺³ = 5⁹ [since aⁿ × a^m = a^m+n]

On equating the coefficients:
2n + 3 = 9
⇒ 2n = 9 − 3
⇒ 2n = 6
∴ n =$\frac{6}{2}=3$

(ii) 8 × 2ⁿ⁺² = 32
⇒ (2)³ × 2ⁿ⁺² = (2)⁵ [since 2³ = 8 and 2⁵ = 32]
⇒ (2)^{3+ (n+2)} = (2)⁵
On equating the coefficients:
3 + n + 2 = 5
⇒ n + 5 = 5
⇒ n = 5 − 5
∴ n = 0

(iii) 6²ⁿ⁺¹ ÷ 36 = 6³
⇒ 6²ⁿ⁺¹ ÷ 6² = 6³ [since 36 = 6²]
⇒ $\frac{6^{2n+1}}{6^2}=6^3$
⇒ $6^{2n+1-2}=6^3$ [since $\frac{a^m}{a^n}=a^{m-n}$ ]
⇒ 6^2n-1 = 6³
On equating the coefficients:
2n - 1 = 3
⇒ 2n = 3 + 1
⇒ 2n = 4
∴ n = $\frac{4}{2}=2$