Question
Simplify :
$\frac{3^5 \times 10^5 \times 25}{5^7 \times 6^5}$
$\frac{3^5 \times 10^5 \times 25}{5^7 \times 6^5}$
✓
Answer
We have, $\frac{3^5 \times 10^5 \times 25}{5^7 \times 6^5}=\frac{3^5 \times(2 \times 5)^5 \times 5^2}{5^7 \times(2 \times 3)^5}$ $\left[\begin{array}{l}\because 10=2 \times 5 \\ 25=5 \times 5=5^2 \\ 6=2 \times 3\end{array}\right]$
$=\frac{3^5 \times 2^5 \times 5^5 \times 5^2}{5^7 \times 2^5 \times 3^5}\left[\because(a \times b)^m=a^m \times b^m\right]$
$=\frac{3^{5-5} \times 2^{5-5} \times 5^{5+2}}{5^7}$ $\left[\begin{array}{c}\because a^m \div a^n=a^{m-n} \\ \text { and } a^m \times a^n=a^{m+n}\end{array}\right]$
$=3^0 \times 2^0 \times 5^{7-7}=3^0 \times 2^0 \times 5^0$
$=1 \times 1 \times 1=1 \quad\left[\because a^0=1\right]$
$=\frac{3^5 \times 2^5 \times 5^5 \times 5^2}{5^7 \times 2^5 \times 3^5}\left[\because(a \times b)^m=a^m \times b^m\right]$
$=\frac{3^{5-5} \times 2^{5-5} \times 5^{5+2}}{5^7}$ $\left[\begin{array}{c}\because a^m \div a^n=a^{m-n} \\ \text { and } a^m \times a^n=a^{m+n}\end{array}\right]$
$=3^0 \times 2^0 \times 5^{7-7}=3^0 \times 2^0 \times 5^0$
$=1 \times 1 \times 1=1 \quad\left[\because a^0=1\right]$
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