Chapter 5.1, Problem 26WE

To determine

To find: Simplify the given expression.

Answer to Problem 26WE

  $\frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=xy$

Explanation of Solution

Given information:

  $\frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}$

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.

  $\frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}$

  $\begin{array}{l}\Rightarrow \frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=\frac{x^{n+1}}{x^n}\cdot \frac{y^n}{y^{n-1}}\\ \Rightarrow \frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=x^{n+1-n}\cdot y^{n-\left(n-1\right)}\\ \Rightarrow \frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=x^{n-n+1}\cdot y^{n-n+1}\\ \Rightarrow \frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=x^1\cdot y^1\\ \Rightarrow \frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=xy\end{array}$

Therefore,

  $\frac{x^{n+1}\cdot y^n}{x^n{y}^{n-1}}=xy$