Q. 129. - Intermediate Algebra

Question

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

$$\begin{gathered} \left(a\right). \left(m^2{n}^2\right){\left(2m{n}^5\right)}^4 \\ \left(b\right). \frac{{\left(-2p^{-2}\right)}^4{\left(3p^4\right)}^2}{{\left(-6p^3\right)}^2} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified form of the expressions

$$\begin{gathered} \left(a\right). {\left(m^{2}n\right)}^2{\left(2m{n}^5\right)}^4=16m^8{n}^{22} \\ \left(b\right). \frac{{\left(-2p^{-2}\right)}^4{\left(3p^4\right)}^2}{{\left(-6p^3\right)}^2}=\frac{4}{p^6} \end{gathered}$$

Part (a) Step 1. Given information.

The given expression is $\left(m^2{n}^2\right){\left(2m{n}^5\right)}^4$.

Part (a) Step 2. Calculate the expression.

Now, apply the product to a power property

We get,

$$\begin{gathered} {\left(m^2\right)}^2.n^2 \times 2^4.m^4.{\left(n^5\right)}^4 \\ =m^4.n^2\times 16.m^4.n^{20} \\ =16m^8{n}^{22} \end{gathered}$$

Part (b) Step 1. Given information.

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

Part (b) Step 2. Calculate the expression.

First of all, apply the product to a power property

We get,

$$\begin{gathered} \frac{-2^4{\left(p^{-2}\right)}^4\times 3^2{\left(p^4\right)}^2}{-6^2\times {\left(p^3\right)}^2} \\ =\frac{16\times 9\times p^{-8}\times p^8}{36p^6} \end{gathered}$$

Now, apply the property of negative exponents and cancel out the common factors

We get,

$4\times \frac{1}{p^6}=\frac{4}{p^6}.$