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In the given question we have to simplify each expression:

$鈸�{(鈭�3y)}^3鈸�{(鈭�8m^2{n}^3)}^0鈸�{(鈭�4x^4)}^{鈭�2}鈸�{\left(2c^{鈭�4}\right)}^3$

Use Power of a Product Property, ${\left(ab\right)}^m=a^m{b}^m$

${(鈭�3y)}^3={{(鈭�3)}^3\left(y\right)}^3$

Simplify.

role="math" localid="1644771786081" ${(鈭�3y)}^3={鈭�27y}^3$

Use Power of a Product Property,${\left(ab\right)}^m=a^m{b}^m$

${(鈭�8m^2{n}^3)}^0={{(鈭�8)}^0{\left(m^2\right)}^0\left(n^3\right)}^0$

Simplify

${(鈭�8m^2{n}^3)}^0=1路1路1$

Multiply.

${(鈭�8m^2{n}^3)}^0=1$

Use Power of a Product Property,${\left(ab\right)}^m=a^m{b}^m$

${(鈭�4x^4)}^{鈭�2}={{(鈭�4)}^{鈭�2}\left(x^4\right)}^{鈭�2}$

Use the Power Property,

localid="1644772244344" $$\begin{gathered} {(鈭�4x^4)}^{鈭�2}={(鈭�4)}^{鈭�2}路x^{4路(鈭�2)} \\ {(鈭�4x^4)}^{鈭�2}={(鈭�4)}^{鈭�2}路x^{鈭�8} \end{gathered}$$

Use the Definition of a negative exponent, $a^{-n}=\frac{1}{a^n}$

${(鈭�4x^4)}^{鈭�2}=\frac{1}{{(鈭�4)}^2}路\frac{1}{x^8}$

Simplify.

${(鈭�4x^4)}^{鈭�2}=\frac{1}{16}路\frac{1}{x^8}$

role="math" localid="1644772776735" ${(鈭�4x^4)}^{鈭�2}=\frac{1}{16x^8}$

Use Power of a Product Property,${\left(ab\right)}^m=a^m{b}^m$

${\left(2c^{鈭�4}\right)}^3={{\left(2\right)}^3\left(c^{鈭�4}\right)}^3$

${\left(2c^{鈭�4}\right)}^3={\left(2\right)}^3{c}^{(鈭�4)路3}$

Simplify.

${\left(2c^{鈭�4}\right)}^3=8c^{鈭�12}$

using, $a^{-n}=\frac{1}{a^n}$

${\left(2c^{鈭�4}\right)}^3=8\frac{1}{c^{12}}$

${\left(2c^{鈭�4}\right)}^3=\frac{8}{c^{12}}$