Problem 53 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{(x-4)^{2}}{x^{2}-16}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{x-4}{x+4}\).

Step 1: Express the Denominator as Factors

Identify and write the denominator \(x^{2}-16\) as difference of squares, i.e., \((x-4)(x+4)\).

Step 2: Break Down the Numerator

Write the numerator \((x-4)^{2}\) explicitly as \((x-4)(x-4)\).

Step 3: Cancel out the Common Terms

Now, cancel out the common factor, \((x-4)\), from the numerator and the denominator. The expression simplifies to \(\frac{x-4}{x+4}\).