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:$\left(−2g^{3}h\right){\left(−3g{j}^4\right)}^2{\left(−ghj\right)}^2$

Apply the laws of indices and simplify.

$$\begin{gathered} \left(−2g^{3}h\right){\left(−3g{j}^4\right)}^2{\left(−ghj\right)}^2=\left(−2g^{3}h\right)\left[{\left(−3\right)}^2{\left(g\right)}^2{\left(j^4\right)}^2\right]\left[{\left(−g\right)}^2{\left(h\right)}^2{\left(j\right)}^2\right] \\ =\left(−2g^{3}h\right)\left[9g^2\left(j^{4×2}\right)\right]\left(g^2{h}^2{j}^2\right) \\ =\left(−2g^{3}h\right)\left[9g^2\left(j^8\right)\right]\left(g^2{h}^2{j}^2\right) \\ =\left(−2g^{3}h\right)\left(9g^2{j}^8\right)\left(g^2{h}^2{j}^2\right) \end{gathered}$$

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

$$\begin{gathered} \left(−2g^{3}h\right){\left(−3g{j}^4\right)}^2{\left(−ghj\right)}^2=\left(−2g^{3}h\right)\left(9g^2{j}^8\right)\left(g^2{h}^2{j}^2\right) \\ =\left(−2×9\right)\left(g^3×g^2×g^2\right)\left(h×h^2\right)\left(j^8×j^2\right) \\ =\left(−18\right)\left(g^{3+2+2}\right)\left(h^{1+2}\right)\left(j^{8+2}\right) \\ =−18g^7{h}^3{j}^{10} \end{gathered}$$