Question

Simplificar raiz de 32 elevado a 5

Answer 1

$( \sqrt{32})^5=(4\sqrt2)^5=1024.4 \sqrt{2}= 4096 \sqrt{2}$

Answer 2

$(\sqrt{32})^5$

Veja que $32=2^5$, então:

$\sqrt{32}=\sqrt{2^5}$.

$\sqrt[c]{a^b}=a^{\frac{b}{c}}$

$\sqrt{32}=\sqrt{2^5}=2^{\frac{5}{2}}$, então:

$(\sqrt{32})^5=\left(2^{\frac{5}{2}}\right)^5$

$(\sqrt{32})^5=2^{\frac{25}{2}}=\sqrt[2]{2^{25}}$

Mas, $2^{25}=(2^{12})^{2}\cdot2$, assim:

$(\sqrt{32})^5=\sqrt[2]{2^{25}}=\sqrt[2]{(2^{12})^{2}\cdot2}$

$(\sqrt{32})^5=2^{12}\cdot\sqrt{2}$

$(\sqrt{32})^5=4~096\sqrt{2}$