Question

Express the following with a single exponent:

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

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

$=3^{15}$………………….$\left(1\right)$

Also, ${\left(3^4\right)}^2$$=3^{4\times 2}$ where $a=3,m=4,n=2$

$=3^8$………………….$\left(2\right)$

Therefore, ${\left(3^3\right)}^5\times {\left(3^4\right)}^2$$=3^{15}\times 3^8$ (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, $3^{15}\times 3^8=3^{15+8}$ where $a=3,m=15,n=8$

$=3^{23}$.

Therefore, ${\left(3^3\right)}^5\times {\left(3^4\right)}^2$$=3^{23}$.