Question

Solve for X
$\sqrt{8 \sqrt{8 \sqrt{8 \sqrt{8.....} } } } } = x$
Correct answer with explanation will be marked as brainliest!​

Answer 1

Given,

$\displaystyle\longrightarrow\sf{x=\sqrt{8\sqrt{8\sqrt{8\sqrt{8\dots\dots}}}}}$

We square both the sides, then,

$\displaystyle\longrightarrow\sf{x^2=8\sqrt{8\sqrt{8\sqrt{8\dots\dots}}}}$

Dividing by 8,

$\displaystyle\longrightarrow\sf{\dfrac{x^2}{8}=\sqrt{8\sqrt{8\sqrt{8\dots\dots}}}}$

That is,

$\displaystyle\longrightarrow\sf{\dfrac{x^2}{8}=x}$

Then,

$\displaystyle\longrightarrow\sf{x^2=8x}$

$\displaystyle\longrightarrow\sf{x^2-8x=0}$

$\displaystyle\longrightarrow\sf{x(x-8)=0}$

$\displaystyle\Longrightarrow\sf{\underline{\underline{x=0\quad;\quad x=8}}}$