Question

If ${4096}^{\frac{x}{4}}=512$, find the value of $x$.

• A. $x=8$
• B. $x=2$
• C. $x=3$
• D. $x=4$

Answer 1

The correct option is C $x=3$

First, we need to simplify the L.H.S of the equation ${4096}^{\frac{x}{4}}=512$.

$4096$ can be expressed as: $4096=8\times 8\times 8\times 8$

Raising the exponent to $\frac{1}{4}$ in both sides of the equation $4096=8\times 8\times 8\times 8$, we get

${4096}^{\frac{1}{4}}=(8\times 8\times 8\times 8{)}^{\frac{1}{4}}$

$\Rightarrow {4096}^{\frac{1}{4}}=(8^4{)}^{\frac{1}{4}}$

$\Rightarrow {4096}^{\frac{1}{4}}=(8{)}^{4\times \frac{1}{4}}$ $[\because (a^m{)}^n=a^{m\times n}]$

$\Rightarrow {4096}^{\frac{1}{4}}=8$

Now, we can say that

${4096}^{\frac{x}{4}}=({4096}^{\frac{1}{4}}{)}^x=8^x$

Now, according to the question,

$8^x=512$ $[\because ({4096}^{\frac{1}{4}}{)}^x=8^x]$

$\Rightarrow 8^x=8\times 8\times 8$ $[\because 512=8\times 8\times 8]$

$\Rightarrow 8^x=8^{(1+1+1)}$ $[\because a^p\times a^q\times a^r=a^{(p+q+r)}]$

$\Rightarrow 8^x=8^3$ [Exponents with equal bases are equal only when powers are equal]

$\Rightarrow x=3$

Thus the value of $x$ in the equation ${4096}^{\frac{x}{4}}=512$ is $3$.

Therefore, option (c.) is the correct answer.