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Chapter 0: Problem 26

Multiply or divide as indicated. $$\frac{x^{2}-4}{x-2} \div \frac{x+2}{4 x-8}$$

Short Answer

Expert verified
The final product of the given expression is 4.

Step by step solution

01

Simplify the Numerator and Denominator

First, factorize both the numerator and denominator. The numerator \(x^{2}-4\) can be factored as \((x-2)(x+2)\) and the the denominator \(4x-8\) can be simplfied as \(4(x-2)\)
02

Rewrite the division as multiplication

The division of fractions can be rewritten as multiplication by flipping the second fraction. So, \(\frac{(x-2)(x+2)}{x-2}\) \(\div\) \(\frac{x+2}{4(x-2)}\) can be rewritten as \(\frac{(x-2)(x+2)}{x-2}\) \(\times\) \(\frac{4(x-2)}{x+2}\)
03

Flip the second fraction

By flipping the second fraction, \(\frac{(x-2)(x+2)}{x-2}\) \(\times\) \(\frac{4(x-2)}{x+2}\) now reads \(\frac{(x-2)(x+2)}{x-2}\) \(\times\) \(\frac{4(x-2)}{x+2}\)
04

Simplify the Final Expression

Now, factorize and cancel out the common terms in the expression. The \(x-2\) term in the numerator of the first fraction can cancel with the \(x-2\) term in the denominator of the first fraction. Similarly, the \(x+2\) term in the denominator of the second fraction can cancel with the \(x+2\) term in the numerator of the first fraction. What is left is the expression \(4\), which is the final answer.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Will help you prepare for the material covered in the next section. If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200\), write a simplified algebraic expression that models the rectangle's perimeter.

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$ 2 x^{4}\left(8 x^{4}+3 x\right) $$

What is a compound inequality and how is it solved?

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\sqrt{20}}{8}=\frac{\sqrt{10}}{4}$$
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