Answer: Option B. -> .06
$$\eqalign{
& {\left( {.000216} \right)^{\frac{1}{3}}} = {\left( {\frac{{216}}{{{{10}^6}}}} \right)^{\frac{1}{3}}} \cr
& = {\left( {\frac{{6 \times 6 \times 6}}{{{{10}^2} \times {{10}^2} \times {{10}^2}}}} \right)^{\frac{1}{3}}} \cr
& = \frac{6}{{{{10}^2}}} \cr
& = \frac{6}{{100}} \cr
& = 0.06 \cr} $$

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

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

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

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

Answer: Option A. -> $$0.\overline 6 $$
$$\eqalign{
& = \sqrt {{\text{0}}{\text{.}}\overline {\text{4}} {\text{ }}} \cr
& = \sqrt {\frac{4}{9}} \cr
& = \frac{2}{3} \cr
& = 0.666...... \cr
& = 0.\overline 6 \cr} $$

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

Answer: Option C. -> 0.63
$$\eqalign{
& = \sqrt {\frac{{0.16}}{{0.4}}} \cr
& = \sqrt {\frac{{0.16}}{{0.40}}} \cr
& = \sqrt {\frac{{16}}{{40}}} \cr
& = \sqrt {\frac{4}{{10}}} \cr
& = \sqrt {0.4} \cr
& = 0.63 \cr} $$

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

Answer: Option D. -> $${{\text{x}}^2} - 1$$
$$\eqalign{
& \Leftrightarrow \sqrt {x + \frac{x}{y}} = x\sqrt {\frac{x}{y}} \cr
& \Leftrightarrow x + \frac{x}{y} = {x^2}.\frac{x}{y} \cr
& \Leftrightarrow \frac{{xy + x}}{y} = \frac{{{x^3}}}{y} \cr
& \Leftrightarrow xy + x = {x^3} \cr
& \Leftrightarrow y + 1 = {x^2} \cr
& \Leftrightarrow y = {x^2} - 1 \cr} $$