Question

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

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

(b) ${\left(\frac{10}{3q}\right)}^{-4}$

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

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

Short answer

(a) $\frac{x^3}{8y^3}$

(b) $\frac{81q^4}{{10}^4}$

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

Part (a) Step 2. Calculate the expression.

To calculate the expression apply the concept of power product.

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

So,

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

Part (b) Step 1. Given information.

The given expression is${\left(\frac{10}{3q}\right)}^{-4}$.

Part (b) Step 2. Calculate the expression.

Apply the concept of negative exponent to the expression.

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

So,

${\left(\frac{10}{3q}\right)}^{-4}={\left(\frac{3q}{10}\right)}^4$

Now apply the power product rule.

${\left(\frac{3q}{10}\right)}^4=\frac{81q^4}{{10}^4}$

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.

Apply the concept of power product.

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

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.

To calculate the expression apply the concept of power rule.

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

So,

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