Question

simplify (5^-1 * 7² )^7/2/ (5² * 7^-4)^7/2

Answer 1

Dear student,

$$\begin{gathered} \frac{{\left(5^{-1}\times 7^2\right)}^{{\displaystyle \frac{7}{2}}}}{{\left(5^2\times 7^{-4}\right)}^{{\displaystyle \frac{7}{2}}}}={\left[\frac{7^2\times 7^4}{5^1\times 5^2}\right]}^{\frac{7}{2}}---(a^{-m}=\frac{1}{a^m}) \\ ={\left(\frac{7^{2+4}}{5^{1+2}}\right)}^{\frac{{\displaystyle 7}}{{\displaystyle 2}}}----(a^m\times a^n=a^{m+n}) \\ ={\left(\frac{7^6}{5^3}\right)}^{\frac{7}{2}}------\left[(a^m{)}^n=a^{mn}\right] \\ =\frac{7^{6\times \frac{7}{2}}}{5^{3\times \frac{7}{2}}}=\frac{7^{21}}{5^{{\displaystyle \frac{21}{2}}}}={\left(\frac{7}{5^{{\displaystyle \frac{-1}{2}}}}\right)}^{21} \end{gathered}$$

Regards