Question

Which of the following is the simplest form of $\frac{9^{\frac{5}{2}}}{4^5}?$

• A. Not possible to simplify
• B. $(\frac{9}{4}{)}^5$
• C. $(\frac{3}{4}{)}^5$
• D. $(\frac{3}{4}{)}^{\frac{25}{2}}$

Answer 1

The correct option is C $(\frac{3}{4}{)}^5$

$\underset{–}{\text{Need to simplify:}}\frac{9^{\frac{5}{2}}}{4^5}$

In the given exponent $\frac{9^{\frac{5}{2}}}{4^5},$ neither bases nor exponents are equal.

Let's try to make exponents/powers equal.
$\frac{9^{\frac{5}{2}}}{4^5}$
$=\frac{(3^2{)}^{\frac{5}{2}}}{4^5}$ [ As $9=3\times 3=3^2$]

$=\frac{3^{2\times \frac{5}{2}}}{4^5}$
[ Repeated exponentiation: $(a^m{)}^n=a^{m\times n}$ ]

$=\frac{3^5}{4^5}$

$\Rightarrow \underset{–}{\frac{9^{\frac{5}{2}}}{4^5}=(\frac{3}{4}{)}^5}$
[ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Division rule of exponent: $\frac{a^n}{b^n}=(\frac{a}{b}{)}^n$ ]

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