91Ó°ÊÓ

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:${\left(x^m\right)}^n=x^{m×n}$

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:$x^m×x^n=x^{m+n}$

The given expression is:${\left(−3d^2{f}^{3}g\right)}^2{\left[{\left(−3d^{2}f\right)}^3\right]}^2$

Apply the laws of indices and simplify.

$$\begin{gathered} {\left(−3d^2{f}^{3}g\right)}^2{\left[{\left(−3d^{2}f\right)}^3\right]}^2=\left[{\left(−3\right)}^2{\left(d^2\right)}^2{\left(f^3\right)}^2{\left(g\right)}^2\right]{\left[{\left(−3\right)}^3{\left(d^2\right)}^3{\left(f\right)}^3\right]}^2 \\ =\left[9\left(d^{2×2}\right)\left(f^{3×2}\right)g^2\right]{\left[−27\left(d^{2×3}\right)f^3\right]}^2 \\ =\left[9\left(d^4\right)\left(f^6\right)g^2\right]{\left[−27\left(d^6\right)f^3\right]}^2 \\ =\left(9d^4{f}^6{g}^2\right){\left(−27d^6{f}^3\right)}^2 \\ =\left(9d^4{f}^6{g}^2\right)\left[{\left(−27\right)}^2{\left(d^6\right)}^2{\left(f^3\right)}^2\right] \\ =\left(9d^4{f}^6{g}^2\right)\left[729\left(d^{6×2}\right)\left(f^{3×2}\right)\right] \\ =\left(9d^4{f}^6{g}^2\right)\left[729\left(d^{12}\right)\left(f^6\right)\right] \\ =\left(9d^4{f}^6{g}^2\right)\left(729d^{12}{f}^6\right) \end{gathered}$$

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

$$\begin{gathered} {\left(−3d^2{f}^{3}g\right)}^2{\left[{\left(−3d^{2}f\right)}^3\right]}^2=\left(9d^4{f}^6{g}^2\right)\left(729d^{12}{f}^6\right) \\ =\left(9×729\right)\left(d^4×d^{12}\right)\left(f^6×f^6\right)\left(g^2\right) \\ =\left(6561\right)\left(d^{4+12}\right)\left(f^{6+6}\right)\left(g^2\right) \\ =6561d^{16}{f}^{12}{g}^2 \end{gathered}$$