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

$$\eqalign{ & = \frac{{3\sqrt 2 }}{{2\sqrt 3 }} \cr & = \frac{{3\sqrt 2 }}{{2\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }} \cr & = \frac{{3\sqrt 6 }}{{2 \times 3}} \cr & = \frac{{\sqrt 6 }}{2} \cr & = \frac{{2.449}}{2} \cr & = 1.2245 \cr} $$

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

$$\eqalign{ & {\text{Clearly, }}a*b = \sqrt {{a^2} + {b^2}} \cr & \therefore 5*12 \cr & = \sqrt {{5^2} + {{12}^2}} \cr & = \sqrt {25 + 144} \cr & = \sqrt {169} \cr & = 13 \cr} $$

L.C.M. of 3, 4, 5, 6, 8 is 120
Now 120 = 2 × 2 × 2 × 3 × 5
To make it a perfect square, it must be multiplied by 2 × 3 × 5
So, required number
$$\eqalign{ & = {2^2} \times {2^2} \times {3^2} \times {5^2} \cr & = 3600 \cr} $$

294 = 7 × 7 × 2 × 3
To make it a perfect square, it must be multiplied by 2 × 3 i.e.,6
∴ Required number = 6

$$\eqalign{ & 0.000326 = \frac{{326}}{{{{10}^6}}} \cr & \,\,\,\,1|\overline 3 \,\,\overline {26} \,\,(18 \cr & \,\,\,\,\,\,\,|\,\,1 \cr & \,\,\,\,\,\,\,| - - - - - - \cr & 28|\,\,\,2\,26 \cr & \,\,\,\,\,\,\,|\,\,\,2\,24 \cr & \,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 \cr} $$
∴ Required number to be subtracted
$$\eqalign{ & = \frac{2}{{{{10}^6}}} \cr & = 0.000002 \cr} $$

$$\eqalign{ & \,\,\,\,\,8|\overline {77} \,\,\overline {00} \,\,(87 \cr & \,\,\,\,\,\,\,\,\,|\,\,64 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - \cr & 167|\,\,\,13\,00 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,11\,69 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,31 \cr & \therefore {\text{Number to be added}} \cr & = {\left( {88} \right)^2} - 7700 \cr & = 7744 - 7700 \cr & = 44 \cr} $$

$$\eqalign{ & \,\,\,\,\,\,\,\,8|\overline {68} \,\,\overline {06} \,\,\overline {21} \,(824 \cr & \,\,\,\,\,\,\,\,\,\,\,\,|\,\,64 \cr & \,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,162|\,\,\,\,4\,06 \cr & \,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,3\,24 \cr & \,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & 1644|\,\,\,\,\,\,\,\,\,\,82\,21 \cr & \,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,65\,76 \cr & \,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,16\,45 \cr & \therefore {\text{Number to be added}} \cr & = {\left( {825} \right)^2} - 680621 \cr & = 680625 - 680621 \cr & = 4 \cr} $$

$$\eqalign{ & \,\,\,\,\,\,\,9|\overline {99} \,\,\overline {99} \,\,(99 \cr & \,\,\,\,\,\,\,\,\,\,|\,81 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - \cr & \,189|\,18\,99 \cr & \,\,\,\,\,\,\,\,\,\,|\,17\,01 \cr & \,\,\,\,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,|\,\,\,\,1\,98 \cr & \therefore {\text{Required number}} \cr & = \left( {9999 - 198} \right) \cr & = 9801 \cr} $$

Least number of 4 digits is 1000
$$\eqalign{ & \,\,\,\,3|\overline {10} \,\,\overline {00} \,\,(31 \cr & \,\,\,\,\,\,\,|\,\,\,\,9 \cr & \,\,\,\,\,\,\,| - - - - - - \cr & 61|\,\,\,\,\,1\,00 \cr & \,\,\,\,\,\,\,|\,\,\,\,\,\,61 \cr & \,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,39 \cr & \therefore {\left( {31} \right)^2} < 1000 < {\left( {32} \right)^2} \cr & {\text{Hence, required number}} \cr & = {\left( {32} \right)^2} \cr & = 1024 \cr} $$