Find the square root of $$\frac{{\left( {0.064 - 0.008} \right)\left( {0.16 - 0.04} \right)}}{{\left( {0.016 + 0.08 + 0.04} \right){{\left( {0.4 + 0.2} \right)}^3}}}$$

Solution (By Examveda Team)

Square root
$$ \Rightarrow \sqrt {\frac{{\left( {0.064 - 0.008} \right)\left( {0.16 - 0.04} \right)}}{{\left( {0.016 + 0.08 + 0.04} \right){{\left( {0.4 + 0.2} \right)}^3}}}} $$
$$ \Rightarrow \sqrt {\frac{{\left[ {{{\left( {0.4} \right)}^3} - {{\left( {0.2} \right)}^3}} \right]\left[ {{{\left( {0.4} \right)}^2} - {{\left( {0.2} \right)}^2}} \right]}}{{\left[ {{{\left( {0.4} \right)}^2} + {{\left( {0.2} \right)}^2} + \left( {0.2 \times 0.4} \right)} \right]{{\left[ {\left( {0.4} \right) + 0.2} \right]}^3}}}} $$
$$ \Rightarrow {\text{Let }}0.4{\text{ }} = {\text{ }}a,{\text{ }}0.2{\text{ }} = {\text{ }}b$$
$$ \Rightarrow \sqrt {\frac{{\left( {{a^3} - {b^3}} \right)\left( {{a^2} - {b^2}} \right)}}{{\left( {{a^2} + {b^2} + ab} \right){{\left( {a + b} \right)}^3}}}} $$
$$ \Rightarrow \sqrt {\frac{{\left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)\left( {a + b} \right)\left( {a - b} \right)}}{{{{\left( {a + b} \right)}^3}\left( {{a^2} + {b^2} + ab} \right)}}} $$
$$\eqalign{ & \Rightarrow \sqrt {\frac{{{{\left( {a - b} \right)}^2}}}{{{{\left( {a + b} \right)}^2}}}} \cr & \Rightarrow \frac{{a - b}}{{a + b}} \cr & \Rightarrow \frac{{0.4 - 0.2}}{{0.4 + 0.2}} \cr & \Rightarrow \frac{{0.2}}{{0.6}} \cr & \Rightarrow \frac{1}{3} \cr} $$