Question

Simplify $\frac{{\left(27\right)}^{\frac{2}{3}}}{{\left(9\right)}^{\frac{1}{2}}{\left(3\right)}^{\frac{3}{2}}}$

• A. √3
• B. 1÷√3
• C. 3√3
• D. 1÷3√3

Answer 1

The correct option is B

1÷√3

Given $\frac{{\left(27\right)}^{\frac{2}{3}}}{{\left(9\right)}^{\frac{1}{2}}{\left(3\right)}^{\frac{3}{2}}}$

Expressing the exponents in terms of 3 ,we get
$(27{)}^{\frac{2}{3}}$ = $(3^3{)}^{\frac{2}{3}}$ = $(3{)}^2$
In the denominator
$9^{\frac{1}{2}}{3}^{\frac{3}{2}}={\left(3^2\right)}^{\frac{1}{2}}{\left(3\right)}^{\frac{3}{2}}={\left(3\right)}^{\frac{5}{2}}$
Hence it is simplified as
$\frac{(3{)}^2}{{\left(3\right)}^{\frac{5}{2}}}$= $(3{)}^{\frac{-1}{2}}$
using ( $a^m÷a^n$ gives us $a^{m-n}$ )
We know (3)${}^{\frac{1}{2}}$ = $\sqrt{3}$
Hence (3)${}^{\frac{-1}{2}}$ = $\frac{1}{\sqrt{3}}$
Rationalising we get
$\frac{\sqrt{3}}{3}$