Question

$\text{If}{\left(\sqrt{4096}\right)}^3=8^x\text{, then}x\text{is}$ ________.

• A. 2
• B. 4
• C. 5
• D. 6

Answer 1

The correct option is D 6

Given:
${\left(\sqrt{4096}\right)}^3=8^x$
Convert 4096 in the form of $2^n$

$\begin{array}{cc}2& 4096\\ 2& 2048\\ 2& 1024\\ 2& 512\\ 2& 256\\ 2& 128\\ 2& 64\\ 2& 32\\ 2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$
$4096=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
= $2^{12}$
${\left(\sqrt{4096}\right)}^3=8^x$

$\Rightarrow \left(\sqrt{(}{2}^{12}\right){)}^3=8^x$.

$\Rightarrow \left(\sqrt{(}\right(2^6{)}^2){)}^3=8^x$. {as ${\left(a^m\right)}^n$=$a^{mn}$}

${\Rightarrow \left(2^6\right)}^3={\left(2^3\right)}^x$

${\Rightarrow \left(2^3\right)}^6={\left(2^3\right)}^x$

$\Rightarrow (8{)}^6$= $(8{)}^x$

We know that, if $a^n$= $a^m$ then, $n=m$.

Hence,
$\Rightarrow x=6$