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Chapter 0: Problem 25

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

Short Answer

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

Step by step solution

01

Rewrite the division as multiplication

Rewrite \(\frac{x^{2}-4}{x} 梅 \frac{x+2}{x-2}\) as \(\frac{x^{2}-4}{x} \times \frac{x-2}{x+2}\), by taking the reciprocal of the second fraction.
02

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 \(\frac{(x-2)(x+2)}{x} \times \frac{x-2}{x+2}\)
03

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 \(\frac{(x-2)}{x}\)
04

Final Answer

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

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