Simplify the expression

${\left(-7a{b}^{4}c\right)}^3{\left[{\left(2a^{2}c\right)}^2\right]}^3$

According to this law,when a product is raised to a power, every factor of the product is raised to the power.

In general:${\left(xy\right)}^m=x^m{y}^m$

If a term with a power is itself raised to a power then the powers are multiplied together.

In general:${\left(x^m\right)}^n=x^{m\times n}$

If the two terms have the same base (in this case x) and are to be multiplied together, then their indices are added.

In general:$x^m\times x^n=x^{m+n}$

The given expression is:${\left(-7a{b}^{4}c\right)}^3{\left[{\left(2a^{2}c\right)}^2\right]}^3$

Apply the laws of indices and simplify.

$$\begin{gathered} {\left(-7a{b}^{4}c\right)}^3{\left[{\left(2a^{2}c\right)}^2\right]}^3=\left[{\left(-7\right)}^3{\left(a\right)}^3{\left(b^4\right)}^3{\left(c\right)}^3\right]{\left[{\left(2\right)}^2{\left(a^2\right)}^2{\left(c\right)}^2\right]}^3 \\ =\left[\left(-343\right)\left(a^3\right)\left(b^{4\times 3}\right){\left(c\right)}^3\right]{\left[4\left(a^{2\times 2}\right){\left(c\right)}^2\right]}^3 \\ =\left(-343a^3{b}^{12}{c}^3\right){\left(4a^4{c}^2\right)}^3 \\ =\left(-343a^3{b}^{12}{c}^3\right)\left[{\left(4\right)}^3{\left(a^4\right)}^3{\left(c^2\right)}^3\right] \\ =\left(-343a^3{b}^{12}{c}^3\right)\left[64\left(a^{4\times 3}\right)\left(c^{2\times 3}\right)\right] \\ =\left(-343a^3{b}^{12}{c}^3\right)\left(64a^{12}{c}^6\right) \end{gathered}$$

Collect the like terms together and use multiplication rule to simplify further.

$$\begin{gathered} {\left(-7a{b}^{4}c\right)}^3{\left[{\left(2a^{2}c\right)}^2\right]}^3=\left(-343a^3{b}^{12}{c}^3\right)\left(64a^{12}{c}^6\right) \\ =\left(-343\times 64\right)\left(a^3\times a^{12}\right)\left(b^{12}\right)\left(c^3\times c^6\right) \\ =\left(-21952\right)\left(a^{3+12}\right)\left(b^{12}\right)\left(c^{3+6}\right) \\ =-21952a^{15}{b}^{12}{c}^9 \end{gathered}$$