Mathematics: Simplify the expression, [tex] ((p^{-2} +rac{1}{p})^{1})^{p} [/tex] , when p=3/4, in both radical and exponents forms.

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Simplify the expression, [tex] ((p^{-2} +\frac{1}{p})^{1})^{p}

Ответ:

thedeathlord123

12.03.2022

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

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

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

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

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Simplify the expression, , when p=3/4, in both radical and exponents forms.

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