Question

In the following exercises, simplify each expression using the quotient to a power property.

(a)${\left(\frac{a}{3b}\right)}^4$

(b) ${\left(\frac{5}{4m}\right)}^{-2}$

(c)${\left(\frac{2x{y}^2}{z}\right)}^3$

(d)${\left(\frac{4p^{-3}}{q^2}\right)}^2$

Short answer

(a)$\frac{a^4}{81b^4}$

(b) $\frac{16m^2}{25}$

(c) $\frac{8x^3{y}^6}{z^3}$

(d) $\frac{16p^{-6}}{q^4}$

Step-by-step solution

Part (a) Step 1. Given information.

The given expression is${\left(\frac{a}{3b}\right)}^4$.

Part (a) Step 2. Calculate the expression.

Applying the concept of power product.

${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$

So,

${\left(\frac{a}{3b}\right)}^4=\frac{a^4}{81b^4}$.

Part (b) Step 1. Given information.

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

Part (b) Step 2. Calculate the expression.

Calculate the expression by applying concept of negative exponent.

So,

${\left(\frac{5}{4m}\right)}^{-2}=\frac{16m^2}{25}$

Part (c) Step 1. Given information.

The given expression is${\left(\frac{2x{y}^2}{z}\right)}^3$.

Part (c) Step 2. Calculate the expression.

Calculate the expression by applying the concept of power product.

So,

$$\begin{gathered} {\left(\frac{2x{y}^2}{z}\right)}^3=\left(\frac{2^3{x}^3{y}^{2·3}}{z^3}\right) \\ =\frac{8x^3{y}^6}{z^3} \end{gathered}$$

Part (d) Step 1. Given information.

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

Part (d) Step 2. Calculate the expression.

The given expression is calculated by applying the concept of power product.

So,

${\left(\frac{4p^{-3}}{q^2}\right)}^2=\frac{16p^{-6}}{q^4}$.