Question

Express the following with a single exponent:

${\left({10}^3\right)}^4\times {\left({10}^5\right)}^3$

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({10}^3\right)}^4$$=$ ${10}^{3\times 4}$ where $a=10,m=3,n=4$

$={10}^{12}$………………….$\left(1\right)$

Also, ${\left({10}^5\right)}^3$ $={10}^{5\times 3}$ where $a=10,m=5,n=3$

$={10}^{15}$………………….$\left(2\right)$

Therefore, ${\left({10}^3\right)}^4\times {\left({10}^5\right)}^3$$={10}^{12}\times {10}^{15}$ (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, ${10}^{12}\times {10}^{15}={10}^{12+15}$ where $a=10,m=12,n=15$

$={10}^{27}$.

Therefore, ${\left({10}^3\right)}^4\times {\left({10}^5\right)}^3$$={10}^{27}$.