Answer:

[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: \: log_{2}[ log_{3}( log_{2}(512) ] = 1\qquad \: \\ \\& \qquad \:\sf \:(2) \: \: \: g = {c}^{(kq)^m} \\ \\& \qquad \:\sf \: (3) \: \: \: {\bigg(x\bigg)}^{ \frac{1}{ {c}^{4}}} \end{aligned}} \qquad \\ \\ [/tex]

Step-by-step explanation:

Given expression is

[tex]\sf \: log_{2}[ log_{3}( log_{2}(512) ] \\ \\ [/tex]

[tex]\sf \: = \: log_{2}[ log_{3}( log_{2}( {2}^{9} ) ] \\ \\ [/tex]

[tex]\sf \: = \: log_{2}[ log_{3}( 9) ] \\ \\ [/tex]

[tex]\qquad\boxed{ \sf{ \: \because \: log_{x}( {x}^{y} ) = y \: }} \\ \\ [/tex]

[tex]\sf \: = \: log_{2}[ log_{3}( {3}^{2} ) ] \\ \\ [/tex]

[tex]\sf \: = \: log_{2}[2] \\ \\ [/tex]

[tex]\sf \: = \: 1 \\ \\ [/tex]

Hence,

[tex]\sf\implies \bf \: log_{2}[ log_{3}( log_{2}(512) ] = 1\\ \\ [/tex]

[tex]\large\underline{\sf{Solution-2}}[/tex]

Given expression is

[tex]\sf \: log_{c}(g) =(kq)^m \\ \\ [/tex]

[tex]\sf\implies \sf \: g = {c}^{(kq)^m} \\ \\ [/tex]

[tex]\qquad\boxed{ \sf{ \: \because \: log_{b}(a) = c \: \: \sf\implies \: a = {b}^{c} \: }} \\ \\ [/tex]

[tex]\large\underline{\sf{Solution-3}}[/tex]

Given expression is

[tex]\sf \: \sqrt[c]{ \sqrt[c]{ \sqrt[c]{ \sqrt[c]{x} } } } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt[c]{ \sqrt[c]{ \sqrt[c]{ {(x)}^{ \frac{1}{c} } } } } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt[c]{ \sqrt[c]{ {(x)}^{ \frac{1}{c} \times \frac{1}{c} } } } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt[c]{ \sqrt[c]{ {(x)}^{ \frac{1}{ {c}^{2} } } } } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt[c]{ {(x)}^{ \frac{1}{ {c}^{2} } \times \frac{1}{c} } } \\ \\ [/tex]

[tex]\sf \: = \: \sqrt[c]{ {(x)}^{ \frac{1}{ {c}^{3} }} } \\ \\ [/tex]

[tex]\sf \: = \: {(x)}^{ \frac{1}{ {c}^{3}} \times \frac{1}{c} } \\ \\ [/tex]

[tex]\sf \: = \: {\bigg(x\bigg)}^{ \frac{1}{ {c}^{4}}} \\ \\ [/tex]

Hence,

[tex]\sf\implies \bf \: \sqrt[c]{ \sqrt[c]{ \sqrt[c]{ \sqrt[c]{x} } } } = {\bigg(x\bigg)}^{ \frac{1}{ {c}^{4}}} \\ \\ [/tex]