Question

Find the value of:
${\left(\frac{a^m}{a^{-n}}\right)}^{m-n}\times {\left(\frac{a^n}{a^{-1}}\right)}^{n-1}\times {\left(\frac{a^1}{a^{-m}}\right)}^{1-m}$

Answer 1

We have ${\left(\frac{a^m}{a^{-n}}\right)}^{m-n}\times {\left(\frac{a^n}{a^{-1}}\right)}^{n-1}\times {\left(\frac{a^1}{a^{-m}}\right)}^{1-m}$

By using law $\frac{a^m}{a^n}=a^{m-n}$

$=(a^{m+n}{)}^{m-n}\times (a^{n+1}{)}^{n-1}\times (a^{m+1}{)}^{1-m}$

$=a^{m^2-n^2}\times a^{n^2-1^2}\times a^{1^2-m^2}$

By using law $a^m\times a^n=a^{m+n}$

$=a^{m^2-n^2+n^2-1+1-m^2}$
$=a^0$
$=1$