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

$$\eqalign{ & {\left( {.000216} \right)^{\frac{1}{3}}} = {\left( {\frac{{216}}{{{{10}^6}}}} \right)^{\frac{1}{3}}} \cr & = {\left( {\frac{{6 \times 6 \times 6}}{{{{10}^2} \times {{10}^2} \times {{10}^2}}}} \right)^{\frac{1}{3}}} \cr & = \frac{6}{{{{10}^2}}} \cr & = \frac{6}{{100}} \cr & = 0.06 \cr} $$

$$\eqalign{ & {\text{Let}}\,\frac{x}{{\sqrt {128} }} = \frac{{\sqrt {162} }}{x} \cr & {\text{Then}}\,{x^2} = \sqrt {128 \times 162} \cr & = \sqrt {64 \times 2 \times 18 \times 9} \cr & = \sqrt {{8^2} \times {6^2} \times {3^2}} \cr & = 8 \times 6 \times 3 \cr & = 144 \cr & \therefore x = \sqrt {144} = 12 \cr} $$

L.C.M. of 21, 36, 66 = 2772.
Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11
To make it a perfect square, it must be multiplied by 7 x 11.
So, required number
= 2² x 3² x 7² x 11²
= 213444

$$\eqalign{ & \,\,\,\,\,\,1|\overline 1 .\overline {56} \overline {25} \,(\,1.25 \cr & \,\,\,\,\,\,\,\,\,|1 \cr & \,\,\,\,\,\,\,\,\,| - - - - - \cr & \,\,\,22|\,\,\,\,56 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,44 \cr & \,\,\,\,\,\,\,\,\,| - - - - - \cr & 245|\,\,\,1225 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,1225 \cr & \,\,\,\,\,\,\,\,\,| - - - - - \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,x \cr & \,\,\,\,\,\,\,\,\,| - - - - - \cr & \therefore \sqrt {1.5625} = 1.25 \cr} $$

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

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

$$\eqalign{ & x = \frac{{ {\sqrt 3 + 1} }}{{ {\sqrt 3 - 1} }} \times \frac{{ {\sqrt 3 + 1} }}{{ {\sqrt 3 + 1} }} \cr & \,\,\,\,\,\, = \frac{{{{\left( {\sqrt 3 + 1} \right)}^2}}}{{ {3 - 1} }} \cr & \,\,\,\,\,\, = \frac{{3 + 1 + 2\sqrt 3 }}{2} \cr & \,\,\,\,\,\, = 2 + \sqrt 3 \cr & y = \frac{{ {\sqrt 3 - 1} }}{{ {\sqrt 3 + 1} }} \times \frac{{ {\sqrt 3 - 1} }}{{ {\sqrt 3 - 1} }} \cr & \,\,\,\,\,\, = \frac{{{{\left( {\sqrt 3 - 1} \right)}^2}}}{{ {3 - 1} }} \cr & \,\,\,\,\,\, = \frac{{3 + 1 - 2\sqrt 3 }}{2} \cr & \,\,\,\,\,\, = 2 - \sqrt 3 \cr & \therefore {x^2} + {y^2} = {\left( {2 + \sqrt 3 } \right)^2} + {\left( {2 - \sqrt 3 } \right)^2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2\left( {4 + 3} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 14 \cr} $$

Money collected = (59.29 x 100) paise = 5929 paise.
∴ Number of members = $$\sqrt {5929} $$   = 77

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

$$\eqalign{ & \frac{{\sqrt 5 }}{2} - \frac{{10}}{{\sqrt 5 }} + \sqrt {125} \, \cr & = \frac{{{{\left( {\sqrt 5 } \right)}^2} - 20 + 2\sqrt 5 \times 5\sqrt 5 }}{{2\sqrt 5 }} \cr & = \frac{{5 - 20 + 50}}{{2\sqrt 5 }} \cr & = \frac{{35}}{{2\sqrt 5 }} \times \frac{{\sqrt 5 }}{{\sqrt 5 }} \cr & = \frac{{35\sqrt 5 }}{{10}} \cr & = \frac{{7 \times 2.236}}{2} \cr & = 7 \times 1.118 \cr & = 7.826 \cr} $$