Square Root And Cube Root - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & a + b + c = 27\sqrt {29} \cr & \Rightarrow 2c + \frac{3}{2}c + c = 27\sqrt {29} \cr & \Rightarrow \frac{9}{2}c = 27\sqrt {29} \cr & \Rightarrow c = 6\sqrt {29} \cr & \therefore \sqrt {{a^2} + {b^2} + {c^2}} \cr & = \sqrt {{{\left( {a + b + c} \right)}^2} - 2\left( {ab + bc + ca} \right)} \cr & = \sqrt {{{\left( {27\sqrt {29} } \right)}^2} - 2\left( {2c \times \frac{3}{2}c + \frac{3}{2}c \times c + c \times 2c} \right)} \cr & = \sqrt {\left( {729 \times 29} \right) - 2\left( {3{c^2} + \frac{3}{2}{c^2} + 2{c^2}} \right)} \cr & = \sqrt {\left( {729 \times 29} \right) - 2 \times \frac{{13}}{2}{c^2}} \cr & = \sqrt {\left( {729 \times 29} \right) - 13{{\left( {6\sqrt {29} } \right)}^2}} \cr & = \sqrt {29\left( {729 - 468} \right)} \cr & = \sqrt {29 \times 261} \cr & = \sqrt {29 \times 29 \times 9} \cr & = 29 \times 3 \cr & = 87 \cr} $$

$$\eqalign{ & = \sqrt {{\text{0}}{\text{.}}\overline {\text{4}} {\text{ }}} \cr & = \sqrt {\frac{4}{9}} \cr & = \frac{2}{3} \cr & = 0.666...... \cr & = 0.\overline 6 \cr} $$

$$\eqalign{ & \,\,\,\,\,4|0\,.\,\overline {20} \,\,\overline {00} \,\,\overline {00} \,\,(.447 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,16 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - - - \cr & \,\,\,84|\,\,\,\,\,\,\,\,\,\,4\,00 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,3\,36 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - - - \cr & \,887|\,\,\,\,\,\,\,\,\,\,\,\,\,\,64\,00 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,62\,09 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - - - \cr & \therefore \sqrt {0.2} = 0.447 \cr} $$

$$\eqalign{ & \,\,\,\,\,3|0\,.\,\overline {12} \,\,\overline {10} \,\,\overline {00} \,\,(0.347 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,9 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - - - \cr & \,\,\,64|\,\,\,\,\,\,\,\,\,\,3\,10 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,2\,56 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - - - \cr & \,687|\,\,\,\,\,\,\,\,\,\,\,\,\,\,54\,00 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,48\,00 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - - - \cr & \therefore \sqrt {0.121} = 0.347 \cr} $$

$$\eqalign{ & = \sqrt {\frac{{0.16}}{{0.4}}} \cr & = \sqrt {\frac{{0.16}}{{0.40}}} \cr & = \sqrt {\frac{{16}}{{40}}} \cr & = \sqrt {\frac{4}{{10}}} \cr & = \sqrt {0.4} \cr & = 0.63 \cr} $$

$$\eqalign{ & = \frac{{1 + \sqrt {0.01} }}{{1 - \sqrt {0.1} }} \cr & = \frac{{1 + 0.1}}{{1 - 0.316}} \cr & = \frac{{1.1}}{{0.684}} \cr & = \frac{{1100}}{{684}} \cr & = 1.6 \cr & \,\,\,\,\,\,\,3|\overline 0 \,.\,\overline {10} \,\,\overline {00} \,\,\overline {00} \,(0.316 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,9 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,61|\,\,\,\,\,\,\,\,\,\,100 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,61 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & 626\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,3900 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,3756 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr} $$

$$\eqalign{ & = \sqrt {{\text{535}}{\text{.9225}}} \cr & = \sqrt {\frac{{{\text{5359225}}}}{{10000}}} \cr & = \frac{{2315}}{{100}} \cr & = 23.15 \cr & \,\,\,\,\,\,\,\,\,2|\overline 5 \,\,\overline {35} \,\,\overline {92} \,\,\overline {25} \,\,(2315 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|4 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,\,43|\,\,1\,35 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,1\,2\,9 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,461\,|\,\,\,\,\,\,\,\,\,6\,92 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,4\,61 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & 4625\,|\,\,\,\,\,\,\,\,\,\,\,\,231\,25 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,231\,25 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{x}} \cr} $$

$$\eqalign{ & {\text{Let the missing digit be x}} \cr & \,\,\,\,\,\,\,1|\overline 1 \,\,\overline {53} \,\,\overline {7{\text{x}}} \,\,(124 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,1 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,22|\,\,\,\,\,\,\,53 \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,44 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & 244\,|\,\,\,\,\,\,\,\,\,\,\,\,\,97{\text{x}} \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,976 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{x}} \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & {\text{Then, x}} = 6 \cr} $$

$$\eqalign{ & = \frac{1}{{\sqrt 5 }} \cr & = \frac{1}{{\sqrt 5 }} \times \frac{{\sqrt 5 }}{{\sqrt 5 }} \cr & = \frac{{\sqrt 5 }}{5} \cr & = \frac{{2.236}}{5} \cr & = 0.447 \cr} $$

$$\eqalign{ & = \sqrt {\frac{8}{3}} \cr & = \sqrt {\frac{{8 \times 3}}{{3 \times 3}}} \cr & = \frac{{\sqrt {24} }}{3} \cr & = \frac{{4.899}}{3} \cr & = 1.633 \cr} $$