Question

Which of the following is equal to $\frac{8^3}{4^2}?$

• A. $2$
• B. $2^3$
• C. $2^4$
• D. $2^5$

Answer 1

The correct option is D $2^5$

$\underset{–}{\text{Need to simplify:}}\frac{8^3}{4^2}$

In the given exponent $\frac{8^3}{4^2},$ neither bases $(8,4)$ nor exponents $(3,2)$ are equal.

Let's try to make bases equal.
$\frac{8^3}{4^2}$
$=\frac{(2^3{)}^3}{(2^2{)}^2}$ [ As $8=2\times 2\times 2=2^3$ and $4=2\times 2=2^2$ ]

$=\frac{2^{3\times 3}}{2^{2\times 2}}$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Repeated exponentiation: $(a^m{)}^n=a^{m\times n}$ ]

$=\frac{2^9}{2^4}$

$=2^{9-4}=2^5$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Division rule of exponents: $\frac{a^m}{a^n}=a^{m-n}$]

$\therefore \underset{–}{\frac{8^3}{4^2}=2^5}$

Therefore, option (d.) is the correct one.