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Chapter 0: Problem 18

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

Short Answer

Expert verified
The simplified form after multiplying and dividing the given polynomials is \( \frac{2}{x-2}\).

Step by step solution

01

Factoring the numerator and denominator of the fractions

Factor the numerator and the denominator of the fractions separately. \(x^{2}-4\) can be factored as \((x-2)(x+2)\); \(x^{2}-4x+4\) as \((x-2)^2\);\(2x-4\) as \(2(x-2)\); \(x+2\) is already a simplified form. Substitute this to have \(\frac{(x-2)(x+2)}{(x-2)^2} \cdot \frac{2(x-2)}{x+2}\)
02

Cancel out common factors

Now, cancel out the common factors that appear in both the numerator and denominator. Here, (x-2) and (x+2) are common and they cancel each other out: \(\frac{(x-2)(x+2)}{(x-2)^2} \cdot \frac{2(x-2)}{x+2}\) simplifies to \(\frac{1}{x-2} \cdot 2\)
03

Final simplification

After the cancelation, we are left with \(\frac{1}{x-2} \cdot 2\) which simplifies to \( \frac{2}{x-2}\)

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