Question

Simplify:

(i) $\frac{25\times t^{-4}}{5^{-3}\times 10\times t^{-8}}$

(ii) $\frac{3^{-5}\times {10}^{-5}\times 125}{5^{-7}\times 6^{-5}}$

Answer 1

(i) $\frac{25\times t^{-4}}{5^{-3}\times 10\times t^{-8}}$

$=\frac{5^2\times t^{-4}}{5^{-3}\times 2\times 5^1\times t^{-8}}$

$=\frac{5^2\times t^{-4}}{5^{(-3+1)}\times 2\times t^{-8}}$ [$\because a^m\times a^n=a^{m+n}$]

$=\frac{5^2\times t^{-4}}{5^{-2}\times 2\times t^{-8}}$

$=\frac{5^{2-(-2)}\times t^{-4-(-8)}}{2}$ [$\because \frac{a^m}{a^n}=a^{m-n},a\ne 0$]

$=\frac{5^{2+2}\times t^{-4+8}}{2}$

$=\frac{5^4\times t^4}{2}$

$=\frac{625t^4}{2}$

$\therefore \frac{25\times t^{-4}}{5^{-3}\times 10\times t^{-8}}=\frac{625t^4}{2}$

(ii)

$\frac{3^{-5}\times {10}^{-5}\times 125}{5^{-7}\times 6^{-5}}$

$=\frac{3^{-5}\times (5\times 2{)}^{-5}\times 5^3}{5^{-7}\times (2\times 3{)}^{-5}}$

$=\frac{3^{-5}\times 2^{-5}\times 5^{-5}\times 5^3}{5^{-7}\times 2^{-5}\times 3^{-5}}$

$=\frac{5^{-5}\times 5^3}{5^{-7}}$

$=\frac{5^{-5+3}}{5^{-7}}$ [Since, $\because a^m\times a^n=a^{m+n}$]

$=\frac{5^{-2}}{5^{-7}}$

$=5^{-2-(-7)}$ [Since, $\because \frac{a^m}{a^n}=a^{m-n},a\ne 0$]

$=5^{-2+7}$

$=5^5$

$\therefore \frac{3^{-5}\times {10}^{-5}\times 125}{5^{-7}\times 6^{-5}}=5^5$