Question

Use the law of exponents and simplify:
$\frac{3^{-4}\times {10}^{-5}\times \left(625\right)}{5^{-3}\times 6^{-4}}$

Answer 1

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

$=\frac{3^{-4}\times {(2\times 5)}^{-5}\times {25}^2}{5^{-3}\times {(3\times 2)}^{-4}}$

$=\frac{3^{-4}\times {(2\times 5)}^{-5}\times 5^4}{5^{-3}\times {(3\times 2)}^{-4}}$

$=\frac{3^{-4}\times 2^{-5}\times 5^{-5}\times 5^4}{5^{-3}\times 3^{-4}\times 2^{-4}}$ ........ [Using product rule of exponents]

$=\frac{3^{(-4+4)}\times 5^{(-5+4+3)}}{2^{(-4+5)}}$ ....... [Using product and quotient rule]

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

$=\frac{25}{2}$

Hence, the answer is $\frac{25}{2}$.