Question

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

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

(b) ${\left(\frac{a}{b}\right)}^{-5}$

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

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

Short answer

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

(b) $\frac{b^5}{a^5}$

(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${\left(\frac{x}{3}\right)}^4$.

Part (a) Step 2. Calculate the expression.

Calculate the expression by applying power product.

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

So,

${\left(\frac{x}{3}\right)}^4=\frac{x^4}{81}$

Part (b) Step 1. Given information.

The given expression is${\left(\frac{a}{b}\right)}^{-5}.$

Part (b) Step 2. Calculate the expression.

Calculate the expression by applying the negative exponent.

${\left(\frac{a}{b}\right)}^{-5}=\frac{b^5}{a^5}$.

Part (c) Step 1. Given expression.

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 power product.

So,

${\left(\frac{2x{y}^2}{z}\right)}^3=\frac{8x^3{y}^6}{z^3}$.

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.

Apply the concept of power product to the expression.

So,

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