Question

Simplify and write $\left[{\left(5^2\right)}^3\times 5^4\right]÷5^7$ as a single exponent.

Answer 1

As we already know for any non-zero integer $\text{'}a\text{'}$ and natural numbers $m$ and $n$,

$$\begin{gathered} {\left(a^m\right)}^n=a^{m\times n}=a^{mn} \\ {a}^m\times a^n=a^{m+n} \\ \frac{a^m}{a^n}=a^{m-n} \end{gathered}$$

On solving now step by step using the above statements:

Firstly, ${\left(5^2\right)}^3=5^{2\times 3}=5^6$ ( using ${\left(a^m\right)}^n=a^{mn}$)

After that, $5^6\times 5^4=5^{6+4}=5^{10}$ (using $a^m\times a^n=a^{m+n}$)

Finally, $\frac{5^{10}}{5^7}=5^{10-7}=5^3$ (using $\frac{a^m}{a^n}=a^{m-n}$)

Combining all the steps: $\left[{\left(5^2\right)}^3\times 5^4\right]÷5^7$$=\left[5^6\times 5^4\right]÷5^7=5^{6+4}÷5^7=5^{10}÷5^7=5^{10-7}=5^3$

Hence, $\left[{\left(5^2\right)}^3\times 5^4\right]÷5^7$ is equal to $5^3$.