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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 result of the multiplication operation is \( \frac{x - 2}{x}\)

Step by step solution

01

Factorize the Numerator

The numerical part in the first fraction \(x^{2}-4\) is a case of difference of squares and thus it can be factored into \((x - 2)(x + 2)\). The fraction hence becomes \(\frac{(x - 2)(x + 2)}{x}\)
02

Change Division to Multiplication

Change the division problem to multiplication by taking the reciprocal of the second fraction. This changes the problem to \(\frac{(x - 2)(x + 2)}{x} \times \frac{x-2}{x+2}\)
03

Cancel out like terms

The term \((x + 2)\) in the numerator of the first fraction cancels out with the term \((x + 2)\) in the denominator of the second fraction. Similarly, \((x - 2)\) in the denominator of the first fraction cancels out with \((x - 2)\) in the numerator of the second fraction. This leaves us with \( \frac{x - 2}{x}\)
04

Final Simplification

The final answer after the multiplication is simplified can't be further simplified and remains as \( \frac{x - 2}{x}\)

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Most popular questions from this chapter

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

Explain how to determine which numbers must be excluded from the domain of a rational expression.
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