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

$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,1|\overline 2 \,.\,\,\overline {00} \,\,\overline {00} \,\,\overline {00} \,(1.414 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|1 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,24|\,\,1\,00 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,96 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,281\,|\,\,\,\,\,\,\,\,\,400 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,281 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & 2824\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11900 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11296 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & \therefore \sqrt 2 = 1.414 \cr} $$

$$\eqalign{ & \Rightarrow 3\sqrt 5 + \sqrt {125} = 17.88 \cr & \Rightarrow {\text{ }}3\sqrt 5 + \sqrt {25 \times 5} = 17.88 \cr & \Rightarrow {\text{ }}3\sqrt 5 + 5\sqrt 5 = 17.88 \cr & \Rightarrow {\text{ }}8\sqrt 5 = 17.88 \cr & \Rightarrow \sqrt 5 = 2.235 \cr & \therefore \sqrt {80} + 6\sqrt 5 \cr & = \sqrt {16 \times 5} + 6\sqrt 5 \cr & = 4\sqrt 5 + 6\sqrt 5 \cr & = 10\sqrt 5 \cr & = \left( {10 \times 2.235} \right) \cr & = 22.35 \cr} $$

Given expression,
$$\sqrt {4 \times 2} + 2\sqrt {16 \times 2} - 3\sqrt {64 \times 2} $$       $$ + $$ $$4\sqrt {25 \times 2} $$
$$\eqalign{ & = 2\sqrt 2 + 8\sqrt 2 - 24\sqrt 2 + 20\sqrt 2 \cr & = 6\sqrt 2 \cr & = 6 \times 1.414 \cr & = 8.484 \cr} $$

$$\eqalign{ & {\text{Given expression,}} \cr & = \frac{{3\sqrt {12} }}{{2\sqrt {28} }} \times \frac{{\sqrt {98} }}{{2\sqrt {21} }} \cr & = \frac{{3\sqrt {4 \times 3} }}{{2\sqrt {4 \times 7} }} \times \frac{{\sqrt {49 \times 2} }}{{2\sqrt {21} }} \cr & = \frac{{6\sqrt 3 }}{{4\sqrt 7 }} \times \frac{{7\sqrt 2 }}{{2\sqrt {21} }} \cr & = \frac{{21\sqrt 6 }}{{4\sqrt {7 \times 21} }} \cr & = \frac{{21\sqrt 6 }}{{28\sqrt 3 }} \cr & = \frac{3}{4}\sqrt 2 \cr & = \frac{3}{4} \times 1.414 \cr & = 3 \times 0.3535 \cr & = 1.0605 \cr} $$

Given expression,
$$ = \sqrt {\frac{{11025}}{{100}}} \times \sqrt {\frac{1}{{100}}} \, \div \,$$    $$\sqrt {\frac{{25}}{{10000}}} \, - \,$$ $$\,\sqrt {\frac{{42025}}{{100}}} $$
$$\eqalign{ & = \frac{{105}}{{10}} \times \frac{1}{{10}} \div \frac{5}{{100}} - \frac{{205}}{{10}} \cr & = \frac{{105}}{{100}} \times \frac{{100}}{5} - \frac{{205}}{{10}} \cr & = 21 - \frac{{205}}{{10}} \cr & = \frac{{210 - 205}}{{10}} \cr & = \frac{5}{{10}} \cr & = \frac{1}{2} \cr & = 0.50 \cr} $$

Sum of decimal places in the numerator and denominator under the radical sign being the same, we remove the decimal.
∴ Given expression,
$$\eqalign{ & = \sqrt {\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}} \cr & = \sqrt {\frac{{81 \times 484}}{{64 \times 625}}} \cr & = \frac{{9 \times 22}}{{8 \times 25}} \cr & = 0.99 \cr} $$

$$\eqalign{ & {\text{Given expression,}} \cr & = \sqrt {\frac{{{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}}}{{{{\left( {\frac{{0.03}}{{10}}} \right)}^2} + {{\left( {\frac{{0.21}}{{10}}} \right)}^2} + {{\left( {\frac{{0.065}}{{10}}} \right)}^2}}}} \cr & = \sqrt {\frac{{100\left[ {{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}} \right]}}{{{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}}}} \cr & = \sqrt {100} \cr & = 10 \cr} $$

$$\eqalign{ & = \sqrt {\left( {7 + 3\sqrt 5 } \right)\left( {7 - 3\sqrt 5 } \right)} \cr & = \sqrt {{{\left( 7 \right)}^2} - {{\left( {3\sqrt 5 } \right)}^2}} \cr & = \sqrt {49 - 45} \cr & = \sqrt 4 \cr & = 2 \cr} $$

$$\eqalign{ & = {\left( {\sqrt 3 - \frac{1}{{\sqrt 3 }}} \right)^2} \cr & = {\left( {\sqrt 3 } \right)^2} + {\left( {\frac{1}{{\sqrt 3 }}} \right)^2} - 2 \times \sqrt 3 \times \frac{1}{{\sqrt 3 }} \cr & = 3 + \frac{1}{3} - 2 \cr & = 1 + \frac{1}{3} \cr & = \frac{4}{3} \cr} $$

$$\eqalign{ & = \sqrt {4{a^2} - 4a + 1} + 3a \cr & = \sqrt {{{\left( 1 \right)}^2} + {{\left( {2a} \right)}^2} - 2 \times 1 \times 2a} + 3a \cr & = \sqrt {{{\left( {1 - 2a} \right)}^2}} + 3a \cr & = \left( {1 - 2a} \right) + 3a \cr & = \left( {1 + a} \right) \cr & = \left( {1 + 0.1039} \right) \cr & = 1.1039 \cr} $$