Question

The real number $(\sqrt[3]{3\sqrt{75}-\sqrt{12}}{)}^{-2}$ when expressed in the simplest form is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{5}$

Answer 1

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

Real number $(\sqrt[3]{3\sqrt{75}-\sqrt{12}}{)}^{-2}$

$\sqrt{75}=5\sqrt{3}$ and $\sqrt{12}=2\sqrt{3}$

$=(\sqrt[3]{35\sqrt{3}-2\sqrt{3}}{)}^{-2}$

$=(\sqrt[3]{33\sqrt{3}}{)}^{-2}$

$(3\sqrt{3}{)}^{\frac{-2}{3}}.......(1)$

$3\sqrt{3}=3^{\frac{1}{2}+1}=3^{\frac{3}{2}}.......\left(ii\right)$

Substituting $\left(ii\right)$ in $\left(i\right)$

${\left[\right(3{)}^{\frac{3}{2}}]}^{\frac{-2}{3}}$

$=\frac{1}{3}=(3{)}^{-1}$