Square Root And Cube Root — Part 2 MCQs with Explanations

$$\eqalign{ & = \sqrt {0.000441} \cr & = \sqrt {\frac{{441}}{{{{10}^6}}}} \cr & = \frac{{\sqrt {441} }}{{\sqrt {{{10}^6}} }} \cr & = \frac{{21}}{{{{10}^3}}} \cr & = \frac{{21}}{{1000}} \cr & = 0.021 \cr} $$

$$\eqalign{ & = {1.5^2} \times \sqrt {0.0225} \cr & = {1.5^2} \times \sqrt {\frac{{225}}{{10000}}} \cr & = 2.25 \times \frac{{15}}{{100}} \cr & = 2.25 \times 0.15 \cr & = 0.3375 \cr} $$

Given expression, $$ = \sqrt {\frac{1}{{100}}} + \sqrt {\frac{{81}}{{100}}} + \sqrt {\frac{{121}}{{100}}} + $$ $$\sqrt {\frac{9}{{10000}}} $$ $$\eqalign{ & = \frac{1}{{10}} + \frac{9}{{10}} + \frac{{11}}{{10}} + \frac{3}{{100}} \cr & = 0.1 + 0.9 + 1.1 + 0.03 \cr & = 2.13 \cr} $$

$$\eqalign{ & {\text{Given expression,}} \cr & \sqrt {1.30} + \sqrt {1300} + \sqrt {0.0130} \cr & = \sqrt {\frac{{130}}{{100}}} + \sqrt {13 \times 100} + \sqrt {\frac{{130}}{{10000}}} \cr & = \frac{{\sqrt {130} }}{{10}} + \sqrt {13} \times 10 + \frac{{\sqrt {130} }}{{100}} \cr & = \frac{{11.40}}{{10}} + 3.605 \times 10 + \frac{{11.40}}{{100}} \cr & = 1.14 + 36.05 + 0.114 \cr & = 37.304 \cr} $$

$$\eqalign{ & \Leftrightarrow \frac{{52}}{x}{\text{ = }}\sqrt {\frac{{169}}{{289}}} \cr & \Leftrightarrow \frac{{52}}{x} = \frac{{13}}{{17}} \cr & \Leftrightarrow x = \left( {\frac{{52 \times 17}}{{13}}} \right) \cr & \Leftrightarrow x = 68 \cr} $$

$$\eqalign{ & {\text{Method 1:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow x = \sqrt {15 \times \left(15 \times 9 \right)} \cr & \Leftrightarrow x = \sqrt {{{15}^2} \times {3^2}} \cr & \Leftrightarrow x = 15 \times 3 \cr & \Leftrightarrow x = 45 \cr & \cr & {\text{Method 2:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow {x^2} = 2025 \cr & \Leftrightarrow x = \sqrt {{2025} } \cr & \Leftrightarrow x = 45 \cr} $$

$$\eqalign{ & {\text{Let }}\frac{x}{{1776}} = \frac{{111}}{x} \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 111 \times 1776 \cr & \Leftrightarrow {x^2} = 111 \times 111 \times 16 \cr & \Leftrightarrow x = \sqrt {{{\left( {111} \right)}^2} \times {{\left( 4 \right)}^2}} \cr & \Leftrightarrow x = 111 \times 4 \cr & \Leftrightarrow x = 444 \cr} $$

$$\eqalign{ & {\text{Let,}} \cr & {\text{ }}\frac{{4\frac{1}{2}}}{x} = \frac{x}{{32}} \cr & {\text{Then,}} \cr & \Leftrightarrow {x^2} = 32 \times \frac{9}{2} \cr & \Leftrightarrow {x^2} = 144 \cr & \Leftrightarrow x = \sqrt {144} \cr & \Leftrightarrow x = 12 \cr} $$

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

$$\eqalign{ & \Leftrightarrow \sqrt {x + \frac{x}{y}} = x\sqrt {\frac{x}{y}} \cr & \Leftrightarrow x + \frac{x}{y} = {x^2}.\frac{x}{y} \cr & \Leftrightarrow \frac{{xy + x}}{y} = \frac{{{x^3}}}{y} \cr & \Leftrightarrow xy + x = {x^3} \cr & \Leftrightarrow y + 1 = {x^2} \cr & \Leftrightarrow y = {x^2} - 1 \cr} $$

$$\eqalign{ & \Leftrightarrow {{\text{n}}^2} = {\left( {{\text{25}}} \right)^{64}} \times {\left( {64} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {\left( {{{\text{5}}^2}} \right)^{64}} \times {\left( {{2^6}} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{150}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{128}} \times {2^{22}} \cr & \Leftrightarrow n = {5^{64}} \times {2^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {\left( {5 \times 2} \right)^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {10^{64}} \times 2048 \cr} $$ ∴ Sum of digits of n = 2 + 0 + 4 + 8 = 14

$$\eqalign{ & \Leftrightarrow \frac{{0.13}}{{{p^2}}} = 13 \cr & \Leftrightarrow {p^2} = \frac{{0.13}}{{13}} \cr & \Leftrightarrow {p^2} = \frac{1}{{100}} \cr & \Leftrightarrow p = \sqrt {\frac{1}{{100}}} \cr & \Leftrightarrow p = \frac{1}{{10}} \cr & \Leftrightarrow p = 0.1 \cr} $$

$$\eqalign{ & \Leftrightarrow \sqrt {{3^n}} = 729 \cr & \Leftrightarrow \sqrt {{3^n}} = {3^6} \cr & \Leftrightarrow {\left( {\sqrt {{3^n}} } \right)^2} = {\left( {{3^6}} \right)^2} \cr & \Leftrightarrow {3^n} = {3^{12}} \cr & \Leftrightarrow n = 12 \cr} $$

$$\eqalign{ & \Leftrightarrow \frac{{\sqrt x }}{{\sqrt {441} }} = 0.02 \cr & \Leftrightarrow \frac{{\sqrt x }}{{21}} = 0.02 \cr & \Leftrightarrow \sqrt x = 0.02 \times 21 \cr & \Leftrightarrow \sqrt x = 0.42 \cr & \Leftrightarrow x = {\left( {0.42} \right)^2} \cr & \Leftrightarrow x = 0.1764{\text{ }} \cr} $$

$$\eqalign{ & {\text{Let ,}} \cr & \sqrt {\frac{{0.0196}}{x}} = 0.2 \cr & {\text{Then,}} \cr & \Leftrightarrow \frac{{0.0196}}{x} = 0.04 \cr & \Leftrightarrow x = \frac{{0.0196}}{{0.04}} \cr & \Leftrightarrow x = \frac{{1.96}}{4} \cr & \Leftrightarrow x = 0.49 \cr} $$

$$\eqalign{ & \Leftrightarrow 37 + \sqrt {0.0615 + x} = 37.25 \cr & \Leftrightarrow \sqrt {0.0615 + x} = 0.25 \cr & \Leftrightarrow 0.0615 + x = {\left( {0.25} \right)^2} = 0.0625 \cr & \Leftrightarrow x = 0.001 \cr & \Leftrightarrow x = \frac{1}{{{{10}^3}}} \cr & \Leftrightarrow x = {10^{ - 3}} \cr} $$

$$\eqalign{ & \Leftrightarrow \sqrt {\left( {x - 1} \right)\left( {y + 2} \right)} = 7 \cr & \Leftrightarrow \left( {x - 1} \right)\left( {y + 2} \right) = {\left( 7 \right)^2} \cr & \Leftrightarrow \left( {x - 1} \right) = 7{\text{ and}}\left( {y + 2} \right) = 7 \cr & \Leftrightarrow x = 8{\text{ and }}y = 5 \cr} $$

$$\eqalign{ & {\text{Let the number be }}x \cr & {\text{Then,}} \cr & \Leftrightarrow \frac{3}{5}{x^2} = 126.15 \cr & \Leftrightarrow {x^2} = \left( {126.15 \times \frac{5}{3}} \right) \cr & \Leftrightarrow {x^2} = 210.25 \cr & \Leftrightarrow x = \sqrt {210.25} \cr & \Leftrightarrow x = 14.5 \cr} $$

$$\eqalign{ & \Rightarrow \sqrt {1 + \frac{x}{{169}}} = \frac{{14}}{{13}} \cr & \Rightarrow 1 + \frac{x}{{169}} = \frac{{196}}{{169}} \cr & \Rightarrow \frac{x}{{169}} = \left( {\frac{{196}}{{169}} - 1} \right) \cr & \Rightarrow \frac{x}{{169}} = \frac{{27}}{{169}} \cr & \Rightarrow x = 27 \cr} $$

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

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

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, $$\eqalign{ & = \sqrt {\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}} \cr & = \sqrt {\frac{{81 \times 484}}{{64 \times 625}}} \cr & = \frac{{9 \times 22}}{{8 \times 25}} \cr & = 0.99 \cr} $$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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