Question

if 8^x+3 = 1022 + 8^x, then find the value of (12x)^1/4

Answer 1

$\textbf{Given:}$

$8^{x+3}=1022+8^x$

$\textbf{To find:}$

$\text{The value of x}$

$\text{Consider}$

$8^{x+3}=1022+8^x$

$\implies\,8^{x+3}-8^x=1022$

$\implies\,8^x(8^3-1)=1022$

$\implies\,8^x(512-1)=1022$

$\implies\,8^x(511)=1022$

$\implies\,8^x=2$

$\implies\,(2^3)^x=2^1$

$\implies\,2^{3x}=2^1$

$\text{Equating powers on bothsides, we get}$

$3x=1$

$\implies\boxed{\bf\,x=\dfrac{1}{3}}$

$\text{Now,}$

$\bf(12x)^{\frac{1}{4}}$

$=(12(\dfrac{1}{3}))^{\frac{1}{4}}$

$=(4)^{\frac{1}{4}}$

$=((\sqrt{2})^4)^{\frac{1}{4}}$

$=\sqrt{2}$

$\therefore\textbf{The value of \bf(12x)^{\frac{1}{4}} is \bf\sqrt2 }$

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