Question

determine o valor de cada letra.
${ \sqrt[12]{9} }^{4} = { \sqrt[3]{9} }^{a}$
${ \sqrt[3]{7} }^{5} = { \sqrt[b]{7}}^{20}$
${ \sqrt[c]{0.6} }^{2} = { \sqrt[16]{0.6} }^{8}$
${ \sqrt[3]{( - 5)} }^{d} = { \sqrt[9]{( - 5)} }^{6}$

Answer 1

Propriedade Utilizada:

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

a)

$\sqrt[12]{9}^4=\sqrt[3]{9}^a\\\\\\9^{\frac{4}{12}}=9^{\frac{a}{3}}\\\\\\\frac{4}{12}=\frac{a}{3}\\\\\\12a=4~.~3\\\\\\a=\frac{12}{12}\\\\\\\boxed{a=1}$

b)

$\sqrt[3]{7}^5=\sqrt[b]{7}^{20}\\\\\\7^{\frac{5}{3}}=7^{\frac{20}{b}}\\\\\\\frac{5}{3}=\frac{20}{b}\\\\\\5b=3~.~20\\\\\\b=\frac{60}{5}\\\\\\\boxed{b=12}$

c)

$\sqrt[c]{0,6}^2=\sqrt[16]{0,6}^8\\\\\\0,6^{\frac{2}{c}}=0,6^{\frac{8}{16}}\\\\\\\frac{2}{c}=\frac{8}{16}\\\\\\8c=2~.~16\\\\\\c=\frac{32}{8}\\\\\\\boxed{c=4}$

d)

$\sqrt[3]{-5}^d=\sqrt[9]{-5}^6\\\\\\(-5)^{\frac{d}{3}}=(-5)^{\frac{6}{9}}\\\\\\\frac{d}{3}=\frac{6}{9}\\\\\\9d=3~.~6\\\\\\d=\frac{18}{9}\\\\\\\boxed{d=2}$