Chapter 5.1, Problem 27WE

To determine

To find: Simplify the given expression.

Answer to Problem 27WE

  $\frac{{\left(pq\right)}^n}{p{q}^n}=p^{n-1}$

Explanation of Solution

Given information:

  $\frac{{\left(pq\right)}^n}{p{q}^n}$

Concept:

  $\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

Using law of exponents

  $\begin{array}{l}\frac{a^m}{a^n}=a^{m-n}\\ a^m\cdot a^n=a^{m+n}\end{array}$

  ${\left(a^n\right)}^m=a^{mn}$

Where $a$ be any value.

  $\frac{{\left(pq\right)}^n}{p{q}^n}$

  $\begin{array}{l}\Rightarrow \frac{{\left(pq\right)}^n}{p{q}^n}=\frac{p^n\cdot q^n}{p\cdot q^n}\\ \Rightarrow \frac{{\left(pq\right)}^n}{p{q}^n}=\frac{p^n}{p}\cdot \frac{q^n}{q^n}\\ \Rightarrow \frac{{\left(pq\right)}^n}{p{q}^n}=p^{n-1}\cdot q^{n-n}\\ \Rightarrow \frac{{\left(pq\right)}^n}{p{q}^n}=p^{n-1}\cdot q^0\\ \Rightarrow \frac{{\left(pq\right)}^n}{p{q}^n}=p^{n-1}\end{array}$

Therefore,

  $\frac{{\left(pq\right)}^n}{p{q}^n}=p^{n-1}$