Question

If 2ⁿ⁻⁷ × 5ⁿ⁻⁴ = 1250, find the value on n.

Answer 1

$2^{n-7}\times 5^{n-4}=1250$
⇒ $\frac{2^n}{2^7}\times \frac{5^n}{5^4}=2\times 5^4$ [since 1250 = 2 × 5⁴]
⇒ $\frac{2^{\mathrm{n}}\times 5^{\mathrm{n}}}{2^7\times 5^4}=2\times 5^4$
⇒ $2^n\times 5^n=2\times 5^4\times 2^7\times 5^4$ [using cross multiplication]
⇒ $2^n\times 5^n=2^{1+7}\times 5^{4+4}$ [since a^m × aⁿ = a^m+n ]
⇒ $2^n\times 5^n=2^8\times 5^8$
⇒ ${\left(2\times 5\right)}^{\mathrm{n}}={\left(2\times 5\right)}^8$ [since aⁿ × bⁿ = (a × b)ⁿ ]
⇒ ${10}^{\mathrm{n}}={10}^8$
⇒ n = 8