Question

Express the following with a single exponent:

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

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

$=2^{12}$………………….$\left(1\right)$

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

$=2^6$………………….$\left(2\right)$

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

$=2^8$………………….$\left(3\right)$

Therefore, ${\left(2^4\right)}^3\times {\left(2^3\right)}^2\times {\left(2^2\right)}^4$$=2^{12}\times 2^6\times 2^8$ (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, $2^{12}\times 2^6\times 2^8=2^{12+6+8}$

$=2^{26}$ .

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