Problem 25 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{2}-4}{x} \div \frac{x+2}{x-2}$$

Step-by-Step Solution

Answer

The final simplified form is \(\frac{(x-2)}{x}\)

Step 1: Rewrite the division as multiplication

Rewrite \(\frac{x^{2}-4}{x} ÷ \frac{x+2}{x-2}\) as \(\frac{x^{2}-4}{x} \times \frac{x-2}{x+2}\), by taking the reciprocal of the second fraction.

Step 2: Factor the expressions

Next, the numerator \(x^{2}-4\) will be factored. \(x^{2}-4\) is a difference of two squares, which can be factored as \((x-2)(x+2)\). So the new expression becomes \(\frac{(x-2)(x+2)}{x} \times \frac{x-2}{x+2}\)

Step 3: Cancel out identical factors

Now, cancel factors that appear in the numerator and the denominator. Here, we can see \((x+2)\) and \((x-2)\) in both, so the expression simplifies to \(\frac{(x-2)}{x}\)

Step 4: Final Answer

So the final simplified form is \(\frac{(x-2)}{x}\)