Question

In the following exercises, simplify each expression by applying several properties.

$$\begin{gathered} \left(a\right).{\left(3p{q}^4\right)}^2{\left(6p^{6}q\right)}^2 \\ \left(b\right).\frac{{\left(-2k^{-3}\right)}^2{\left(6k^2\right)}^4}{{\left(9k^4\right)}^2} \end{gathered}$$

Short answer

The simplified form of the expressions

$$\begin{gathered} \left(a\right).{\left(3p{q}^4\right)}^2{\left(6p^{6}q\right)}^2=324p^{14}{q}^{10} \\ \left(b\right).\frac{{\left(-2k^{-3}\right)}^2{\left(6k^2\right)}^4}{{\left(9k^4\right)}^2}=\frac{64}{k^6} \end{gathered}$$

Step-by-step solution

Part (a) Step 1. Given information.

The given expression is ${\left(3p{q}^4\right)}^2{\left(6p^{6}q\right)}^2$.

Part (a) Step 2. Apply the product to a power property. 

Apply the concept of product power-

$$\begin{gathered} {\left(3pq4\right)}^2{\left(6p6q\right)}^2=3^2.p^2.{\left(q^4\right)}^2\times 6^2.{\left(p^6\right)}^2.q^2 \\ =9p^2{q}^{8}36p^{12}{q}^2 \\ =324p^{14}{q}^{10}. \end{gathered}$$

Part (b) Step 1. Given information.

The given expression is$\frac{{(-2k^{-3})}^2{\left(6k^2\right)}^4}{{\left(9k^4\right)}^2}$.

Part (b) Step 2. Apply the concept of power property. 

Apply the power product rule-

$$\begin{gathered} \frac{{(-2k^{-3})}^2{\left(6k^2\right)}^4}{{\left(9k^4\right)}^2}=\frac{-2^2{\left(k^{-3}\right)}^2.6^4.{\left(k^2\right)}^4}{9^2.{\left(k^4\right)}^2} \\ =\frac{4\times 1296\times k^2}{81\times k^8} \\ =\frac{64}{k^6} \end{gathered}$$