Solution:
$$\eqalign{ \sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt 2 } } } } \cr = \sqrt {2\sqrt {2\sqrt {2\sqrt {{{2.2}^{\frac{1}{2}}}} } } } \cr = \sqrt {2\sqrt {2\sqrt {2\sqrt {{2^{\left( {1 + \frac{1}{2}} \right)}}} } } } \cr = \sqrt {2\sqrt {2\sqrt {2\sqrt {{2^{\frac{3}{2}}}} } } } \cr = \sqrt {2\sqrt {2\sqrt {2.{{\left( {{2^{\frac{3}{2}}}} \right)}^{\frac{1}{2}}}} } } \cr = \sqrt {2\sqrt {2\sqrt {{{2.2}^{\left( {\frac{3}{2} \times \frac{1}{2}} \right)}}} } } \cr = \sqrt {2\sqrt {2\sqrt {{{2.2}^{\frac{3}{4}}}} } } \cr = \sqrt {2\sqrt {2\sqrt {{2^{\left( {1 + \frac{3}{4}} \right)}}} } } \cr = \sqrt {2\sqrt {2\sqrt {{2^{\frac{7}{4}}}} } } \cr = \sqrt {2\sqrt {2{{\left( {{2^{\frac{7}{4}}}} \right)}^{\frac{1}{2}}}} } \cr = \sqrt {2\sqrt {{{2.2}^{\left( {\frac{7}{4} \times \frac{1}{2}} \right)}}} } \cr = \sqrt {2\sqrt {{{2.2}^{\frac{7}{8}}}} } \cr = \sqrt {2\sqrt {{2^{\left( {1 + \frac{7}{8}} \right)}}} } \cr = \sqrt {2\sqrt {{2^{\frac{{15}}{8}}}} } \cr = \sqrt {2{{\left( {{2^{\frac{{15}}{8}}}} \right)}^{\frac{1}{2}}}} \cr = \sqrt {{{2.2}^{\frac{{15}}{{16}}}}} \cr = \sqrt {{2^{\left( {1 + \frac{{15}}{{16}}} \right)}}} \cr = \sqrt {{2^{\frac{{31}}{{16}}}}} \cr = {\left( {{2^{\frac{{31}}{{16}}}}} \right)^{\frac{1}{2}}} \cr = {2^{\left( {\frac{{31}}{{16}} \times \frac{1}{2}} \right)}} \cr = {2^{\frac{{31}}{{32}}}} \cr} $$

Solution:
$$\eqalign{ \sqrt {6 - 4\sqrt 3 + \sqrt {16 - 8\sqrt 3 } } \cr = \sqrt {6 - 4\sqrt 3 + \sqrt {12 + 4 - 8\sqrt 3 } } \cr = \sqrt {6 - 4\sqrt 3 + \sqrt {{{\left( {2\sqrt 3 } \right)}^2} + {{\left( 2 \right)}^2} - 2 \times 2\sqrt 3 \times 2} } \cr = \sqrt {6 - 4\sqrt 3 + \sqrt {{{\left( {2\sqrt 3 - 2} \right)}^2}} } \cr = \sqrt {6 - 4\sqrt 3 + 2\sqrt 3 - 2} \cr = \sqrt {{{\left( {\sqrt 3 } \right)}^2} + {{\left( 1 \right)}^2} - 2 \times \sqrt 3 \times 1} \cr = \sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} \cr = \sqrt 3 - 1 \cr} $$

Solution:
$$\eqalign{ {{\text{3}}^{10}} \times {\text{2}}{{\text{7}}^2}{\text{ = }}{{\text{9}}^2} \times {{\text{3}}^n} \cr \Rightarrow {{\text{3}}^{10}} \times {\left( {{3^3}} \right)^2}{\text{ = }}{\left( {{3^2}} \right)^2} \times {{\text{3}}^n} \cr \Rightarrow {{\text{3}}^{10 + 6}}{\text{ = }}{{\text{3}}^{4 + n}} \cr \Rightarrow 10 + 6 = 4 + n \cr \Rightarrow n = 12 \cr} $$

Solution:
$$\eqalign{ \frac{{343 \times 49}}{{216 \times 16 \times 81}} \cr = \frac{{{7^3} \times {7^2}}}{{{6^3} \times {2^4} \times {3^4}}} \cr = \frac{{{7^{\left( {3 + 2} \right)}}}}{{{6^3} \times {{\left( {2 \times 3} \right)}^4}}} \cr = \frac{{{7^5}}}{{{6^3} \times {6^4}}} \cr = \frac{{{7^5}}}{{{6^{\left( {3 + 4} \right)}}}} \cr = \frac{{{7^5}}}{{{6^7}}} \cr} $$

Solution:
a = b^p and b = c^q
∴ c = a^r = (b^p)^r = (b)^pr = (c^q)^pr = c^pqr
⇒ pqr = 1

Solution:
$$\eqalign{ {\left( {32 \times {{10}^{ - 5}}} \right)^{ 2}} \times 64 \div \left( {{2^{16}} \times {{10}^{ - 4}}} \right) = {10^?} \cr \Rightarrow {\left( {{2^5} \times {{10}^{ - 5}}} \right)^{ 2}} \times {2^6} \div \left( {{2^{16}} \times {{10}^{ - 4}}} \right) \cr \,\,\,\,\,\,\,\,\, = {10^?}.....\left[ {\because {{\left( {{a^m}} \right)}^n} = {a^{mn}}} \right] \cr \Rightarrow \frac{{{2^{10}} \times {{10}^{ - 10}} \times {2^6}}}{{{2^{16}} \times {{10}^{ - 4}}}} \cr \,\,\,\,\,\,\,\,\,\, = {10^?}.....\left[ {\because {a^m} \times {a^n} = {a^{m + n}}} \right] \cr \Rightarrow \frac{{{2^{16}} \times {{10}^4}}}{{{2^{16}} \times {{10}^{10}}}} \cr \,\,\,\,\,\,\,\, = {10^?}.....\left[ {{a^{ - m}} = \frac{1}{{{a^m}}}} \right] \cr \Rightarrow {10^{4 - 10}} = {10^?} \cr \Rightarrow {10^{ - 6}} = {10^?} \cr \Rightarrow ? = - 6 \cr} $$

Solution:
$$\eqalign{ \root 3 \of {{2^4}\sqrt {{2^{ - 5}}\sqrt {{2^6}} } } {\text{ }} \cr = \root 3 \of {{2^4}\sqrt {{2^{ - 5}}{{\left( {{2^6}} \right)}^{\frac{1}{2}}}} } {\text{ }} \cr = \root 3 \of {{2^4}\sqrt {{2^{ - 5}}{{\left( 2 \right)}^{\left( {6 \times \frac{1}{2}} \right)}}} } {\text{ }} \cr = \root 3 \of {{2^4}\sqrt {{2^{ - 5}}{{.2}^3}} } {\text{ }} \cr = \root 3 \of {{2^4}\sqrt {{2^{\left( { - 5 + 3} \right)}}} } {\text{ }} \cr = \root 3 \of {{2^4}\sqrt {{2^{ - 2}}} } {\text{ }} \cr = \root 3 \of {{2^4}.{{\left( {{2^2}} \right)}^{\frac{1}{2}}}} \cr {\text{ = }}\root 3 \of {{2^4}{{.2}^{\left( { - 2 \times \frac{1}{2}} \right)}}} \cr = \root 3 \of {{2^4}{{.2}^{\left( { - 1} \right)}}} \cr = \root 3 \of {{2^{\left( {4 - 1} \right)}}} \cr {\text{ = }}\root 3 \of {{2^3}} \cr {\text{ = }}{\left( {{2^3}} \right)^{\frac{1}{3}}} \cr {\text{ = }}{{\text{2}}^{\left( {3 \times \frac{1}{3}} \right)}} \cr {\text{ = 2 }} \cr} $$