Problem 54 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{(x+5)^{2}}{x^{2}-25}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{x+5}{x-5}\).

Step 1: Factorize the Denominator

Notice that our denominator \(x^{2}-25\) is a difference of two squares, hence it can be factored into \((x+5)(x-5)\). So the rational expression becomes \(\frac{(x+5)^{2}}{(x+5)(x-5)}\).

Step 2: Simplify the expression

Observe that the rational expression shares a common term in the numerator and in the denominator. Cancel out the \(x+5\) terms. This gives the final simplified expression \(\frac{x+5}{x-5}\).