Question

Simplify:

$\frac{3^{-4}\times {10}^{-5}\times \left(625\right)}{5^{-3}\times 6^{-4}}$

Answer 1

$\frac{3^{-4}\times {10}^{-5}\times 625}{5^{-3}\times 6^{-4}}$

=$\frac{3^{-4}\times {10}^{-5}\times (5\times 5\times 5\times 5)}{5^{-3}\times 6^{-4}}$

$=\frac{3^{-4}\times {(2\times 5)}^{-5}\times 5^4}{5^{-3}\times {(2\times 3)}^{-4}}$ [$\because a\times a\times a....m\text{times}=a^m,a\ne 0$]

$=\frac{3^{-4}\times 2^{-5}\times 5^{-5}\times 5^4}{5^{-3}\times 2^{-4}\times 3^{-4}}$ [ $\because (ab{)}^m=a^m\times b^m$]

$=3^{-4+4}\times 2^{-5+4}\times 5^{-5+4+3}$ [$\because \frac{a^m}{a^n}=a^{m-n},a\ne 0$]

$=3^0\times 2^{-1}\times 5^2$

$=1\times \frac{1}{2}\times 25$ [$\because a^{-1}=\frac{1}{a},a\ne 0$]

$=\frac{25}{2}$

$\therefore \frac{3^{-4}\times {10}^{-5}\times 625}{5^{-3}\times 6^{-4}}=\frac{25}{2}$