Question

Express the following with a single exponent:

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

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

$=3^6$………………….$\left(1\right)$

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

$=3^6$………………….$\left(2\right)$

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

$=3^{12}$………………….$\left(3\right)$

Therefore, ${\left(3^2\right)}^3\times {\left(3^3\right)}^2\times {\left(3^4\right)}^3$$=3^6\times 3^6\times 3^{12}$ (From $\left(1\right)$, $\left(2\right)$ and $\left(3\right)$)

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

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

So, $3^6\times 3^6\times 3^{12}=3^{6+6+12}$

$=3^{24}$ .

Therefore, ${\left(3^2\right)}^3\times {\left(3^3\right)}^2\times {\left(3^4\right)}^3$$=3^{24}$.