Tag: estimating square roots

Questions Related to estimating square roots

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

maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $\dfrac {1}{2}$

  2. $\dfrac {1}{3}$

  3. $\dfrac {1}{4}$

  4. $\dfrac {1}{5}$

Show answer
Correct Option: B
Explanation:

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

$\sqrt{75}=5\sqrt{3}$ and $\sqrt{12}=2\sqrt{3}$
$=(\sqrt[3]{5\sqrt{3}-2\sqrt{3}})^{-2}$
$=(\sqrt[3]{3\sqrt{3}})^{-2}$
$(3\sqrt{3})^{\dfrac{-2}{3}} ....... (1)$
$3\sqrt{3}=3^{\dfrac{1}{2}+1}=3^{\dfrac{3}{2}} ....... (ii)$
Substituting $(ii)$ in $(i)$
$\left[(3)^{\dfrac{3}{2}}\right]^{\dfrac{-2}{3}}$
$=\dfrac{1}{3}=(3)^{-1}$