Chapter 8.5, Problem 39PPSE

Solution

To state:If the closure property of rational numbers extends to the rational expression. Explain and describe any restrictions on rational expressions.

Yes, the closure property of rational numbers extends to the rational expression.

Given information:

Theclosure property of rational numbers.

Explanation:

To simplify the sum and difference of rational expressions, look for the LCD of the expressions. Then make equivalent expressions using the LCD and combine the terms.

Also notice the restrictions on the variables.

For example, take rational expression $\frac{x-2}{x+4}$ .

One of the rational expressions that simplifies to $\frac{x-2}{x+4}$ is $\frac{\frac{2x-4}{x+1}}{\frac{2x+8}{x+1}}$ .

Rewrite the expression.

  $\frac{\frac{2x-4}{x+1}}{\frac{2x+8}{x+1}}=\frac{2x-4}{x+1}÷\frac{2x+8}{x+1}$

Multiply by the reciprocal.

  $\frac{\frac{2x-4}{x+1}}{\frac{2x+8}{x+1}}=\frac{2x-4}{x+1}\cdot \frac{x+1}{2x+8}$

Factor and note the restrictions.

  $\begin{array}{c}\frac{\frac{2x-4}{x+1}}{\frac{2x+8}{x+1}}=\frac{2\left(x-2\right)}{x+1}\cdot \frac{x+1}{2\left(x+4\right)}\\ =\frac{x-2}{x+4}\\ x\ne -4,-1\end{array}$

As the division of two rational expressions gives and rational expression, the closure property is satisfied.

Thus, theclosure property of rational numbers extends to the rational expression.