Question

Express the following with a single exponent:

${\left(5^3\right)}^6\times {\left(5^2\right)}^4$

Answer 1

For any non-zero integer $a$, where $m\mathrm{and}n$ are whole numbers,

${\left(a^m\right)}^n=a^{m\times n}$

So, ${\left(5^3\right)}^6$$=$ $5^{3\times 6}$ where $a=5,m=3,n=6$

$=5^{18}$………………….$\left(1\right)$

Also, ${\left(5^2\right)}^4$$=5^{2\times 4}$ where $a=5,m=2,n=4$

$=5^8$………………….$\left(2\right)$

Therefore, ${\left(5^3\right)}^6\times {\left(5^2\right)}^4$$=5^{18}\times 5^8$ (From $\left(1\right)$ and $\left(2\right)$)

we know that, $a^m\times a^n=a^{m+n}$

Exponents can be added only when the bases are the same.

So, $5^{18}\times 5^8=5^{18+8}$ where $a=5,m=18,n=8$

$=5^{26}$.

Therefore, ${\left(5^3\right)}^6\times {\left(5^2\right)}^4$$=5^{26}$.