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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 simplified form of the expression is \(x-2\) where \(x\neq 2\) and \(x\neq -2\)

Step by step solution

01

Simplify the rational expressions

Simplify the numerator of the first rational expression using difference of square rule. The expression \(x^{2}-4\) can be written as \((x+2)(x-2)\). Thus the first rational expression becomes \(\frac{(x+2)(x-2)}{x}\)
02

Change to Multiplication

Because dividing is similar to multiplying by the reciprocal, change the operation from division (\(\div\)) to multiplication (\(\times\)) and reverse (or find the reciprocal of) the second fraction. The expression becomes \(\frac{(x+2)(x-2)}{x} \times \frac{(x-2)}{(x+2)}\)
03

Simplify the expression

In rational expressions multiplication or division, any common similarities can be cancelled out from the numerator and denominator. Here, the common terms present are \((x+2)\) and \((x-2)\), which can be cancelled out. The expression simplifies to \(x-2\) when \(x\neq 2\) and \(x\neq -2\).
04

Step 4:State the Restrictions

As a final step, always remember to state the restrictions on the variable. The restrictions come from ensuring that the original denominators are not zero. Therefore state that \(x \neq 2\) and \(x \neq -2\) from the original denominators

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