Question

Simplify the expression,
`((p^(-2) +(1)/(p))^(1))^(p)` , when p=3/4, in both radical and exponents forms.

Answer 1

`\bf \left[ \left( p^(-2)+\cfrac{1}{p} \right)^1 \right]^p\implies \left[ \left( \cfrac{1}{p^2}+\cfrac{1}{p} \right)^1 \right]^p\implies \left[ \left( \cfrac{1+p}{p^2} \right)^1 \right]^p\\\\\\\left( \cfrac{1+p}{p^2} \right)^(1\cdot p)\implies \left( \cfrac{1+p}{p^2} \right)^p\implies \stackrel{p=(3)/(4)}{\left( \cfrac{1+(3)/(4)}{\left( (3)/(4) \right)^2} \right)^{(3)/(4)}}`

`\bf \sqrt[4]{\left( \cfrac{1+(3)/(4)}{\left( (3)/(4) \right)^2} \right)^3}\implies \sqrt[4]{\left( \cfrac{~~(7)/(4)~~}{(9)/(16)} \right)^3}\implies \sqrt[4]{\left( \cfrac{7}{4}\cdot \cfrac{16}{9} \right)^3}\implies \sqrt[4]{\left( \cfrac{28}{9} \right)^3}`

`\bf \sqrt[4]{\cfrac{28^3}{9^3}}\implies \sqrt[4]{\cfrac{21952}{729}}\implies \cfrac{2\sqrt[4]{1372}}{3\sqrt[4]{3}}\\\\\\\cfrac{2}{3}\sqrt[4]{\cfrac{1372}{3}}\implies \stackrel{\textit{exponent form}}{\cfrac{2}{3}\left( \cfrac{1372}{3} \right)^{(1)/(4)}}`