Question

Simplify: $\frac{3^5\times {10}^5\times 25}{5^7\times 6^5}$

Answer 1

Simplify the given expression.

Step 1: Simplify the bases further.

${10}^5$ can be expressed as ${10}^5={\left(2\times 5\right)}^5=2^5\times 5^5$.

$25$ can be expressed as $25=5\times 5=5^2$

$6^5$ can be expressed as $6^5={\left(2\times 3\right)}^5=2^5\times 3^5$.

Step 2: Substitute these values in the given expression and simplify using exponent rules.

$\begin{array}{rcl}\frac{3^5\times {10}^5\times 25}{5^7\times 6^5}& =& \frac{3^5\times 2^5\times 5^5\times 5^2}{5^7\times 2^5\times 3^5}\\ & =& \frac{3^5\times 2^5\times 5^{\left(5+2\right)}}{5^7\times 2^5\times 3^5}\left[\because a^m\times a^n=a^{m+n}\right]\\ & =& \frac{3^5\times 2^5\times 5^7}{5^7\times 2^5\times 3^5}\\ & =& 3^{\left(5-5\right)}\times 2^{\left(5-5\right)}\times 5^{\left(7-7\right)}\left[\because \frac{a^m}{a^n}=a^{m-n}\right]\\ & =& 3^0\times 2^0\times 5^0\\ & =& 1\times 1\times 1\\ & =& 1\end{array}$

Hence, $\frac{3^5\times {10}^5\times 25}{5^7\times 6^5}=1$.