Question

5. Determine O Valor De
$x$
Em Cada Caso.

a)
$\sqrt[12]{ {2}^{8} } = \sqrt[x]{ {2}^{2} }$
b)
$\sqrt[10]{3 {}^{15} } = \sqrt[x]{ {3}^{3} }$
c)
$\sqrt[27]{512} = \sqrt[3]{ {2}^{x} }$
d)
$\sqrt[10]{ \frac{81}{625} } = \sqrt[5]({ \frac{3}{5} }) {}^{x}$
​

Answer 1

A)

$\sqrt[12]{ {2}^{8} } = \sqrt[x]{ {2}^{2} }$

${2}^{ \frac{8}{12} } = {2}^{ \frac{2}{x} }$

• Em uma equação exponencial quando as bases são iguais nois dois membros cortamos elas, sobrando apenas seus expoentes:

$\frac{8}{12} = \frac{2}{x}$

$8x = 12 \times 2$

$8x = 24$

$x = \frac{24}{8}$

$\boxed{\red{x = 3}}$

B)

$\sqrt[10]{3 {}^{15} } = \sqrt[x]{ {3}^{3} }$

${3}^{ \frac{15}{10} } = {3}^{ \frac{3}{x} }$

$\frac{15}{10} = \frac{3}{x}$

$\frac{3}{2} = \frac{3}{x}$

$3x = 6$

$x = \frac{6}{3}$

$\boxed{\red{x = 2}}$

C)

$\sqrt[27]{512} = \sqrt[3]{ {2}^{x} }$

$\sqrt[27]{ {2}^{9} } = \sqrt[3]{ {2}^{x} }$

${2}^{ \frac{9}{27} } = {2}^{ \frac{x}{3} }$

$\frac{9}{27} = \frac{x}{3}$

$\frac{1}{3} = \frac{x}{3}$

$3x = 3$

$x = \frac{3}{3}$

$\boxed{\red{x = 1}}$

D)

$\sqrt[10]{ \frac{81}{625} } = \sqrt[5]({ \frac{3}{5} }) {}^{x}$

$\sqrt[10]{( \frac{3}{5}) ^{4} } = \sqrt[5]{ (\frac{3}{5} ) ^{x} }$

$( \frac{3}{5} ) ^{ \frac{4}{10} } = ( \frac{3}{5} ) ^{ \frac{x}{5} }$

$\frac{4}{10} = \frac{x}{5}$

$\frac{4}{2} = \frac{x}{1}$

$2x = 4$

$x = \frac{4}{2}$

$\boxed{\red{x = 2}}$

Espero ter ajudado!