Question

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

(a) ${\left(\frac{p}{2}\right)}^5$

(b) ${\left(\frac{x}{y}\right)}^{-6}$

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

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

Short answer

(a) $\left(\frac{p^5}{32}\right)$

(b) $\frac{y^6}{x^6}$

(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 information is (p/2)⁵.

Part (a) Step 2. Calculate the expression.

Calculate the expression by power product.

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

So,

${\left(\frac{p}{2}\right)}^5=\frac{p^5}{32}.$

Part (b) Step 1. Given information.

The given expression is${\left(\frac{x}{y}\right)}^{-6}$.

Part (b) Step 2. Calculate the expression.

Calculate the expression by negative exponent.

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

So,

${\left(\frac{x}{y}\right)}^{-6}=\frac{y^6}{x^6}$

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 by applying the power product rule.

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

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.

Calculate the expression by applying power product.

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