Question

If $n$ and $k$ are positive integers and $8^n=2^k$, what is the value of $\frac{n}{k}$?

• A. $\frac{1}{4}$
• B. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $3$

Answer 1

The correct option is B $\frac{1}{3}$

Given

$8^n$ $=$ $2^k$

As ${}^{\prime }{8}^{\prime }$ can be written as ${}^{\prime }$$2^3$${}^{\prime }$,

${\left(2^3\right)}^n$ $=$ $2^k$

$2^{3\times n}$ $=$ $2^k$

$2^{3n}$ $=$ $2^k$

Equating the exponents as the bases are same, we get

$3n$ $=$ $k$

Rearranging the terms, we get

$\frac{n}{k}$ $=$ $\frac{1}{3}$

Therefore, the value of $\frac{n}{k}=\frac{1}{3}$.