Chapter 1.6, Problem 141E

To determine

To simplify: the rational expression $\frac{5x^2{y}^2+25x^{2}y}{xy+5x}$ .

Answer to Problem 141E

  $5xy$ Where $x\ne 0$ and $y\ne -5$

Explanation of Solution

Given:

The rational expression: $\frac{5x^2{y}^2+25x^{2}y}{xy+5x}$

First factor the numerator and the denominator.

  $\frac{5x^2{y}^2+25x^{2}y}{xy+5x}=\frac{5x^{2}y\left(y+5\right)}{x\left(y+5\right)}$

Now, find the values for which the denominator becomes 0, as division by 0 is undefined so for such values the given expression is not defined.

Now, the denominator becomes 0 when either $x=0$ or $y=-5$ .

So, set the restriction on the domain of this expression to be $x\ne 0$ and $y\ne -5$ .

Lastly cancel out the common factor.

  $\begin{array}{c}\frac{5x^2{y}^2+25x^{2}y}{xy+5x}=\frac{5x\cdot \cancel{x}\cdot y\cancel{\left(y+5\right)}}{\cancel{x}\cancel{\left(y+5\right)}}\\ =5xy\end{array}$

Conclusion:

The simplified form of the given rational expression is:

  $5xy$ Where $x\ne 0$ and $y\ne -5$