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Chapter 0: Problem 21

Multiply or divide as indicated. $$\frac{x^{3}-8}{x^{2}-4} \cdot \frac{x+2}{3 x}$$

Short Answer

Expert verified
The simplified expression is \( (x^{2}+2x+4) / (3x) \).

Step by step solution

01

Factorize the expressions

First, factorize where you can. The expression \(x^{3}-8\) can be factored into \((x-2)(x^{2}+2x+4)\) using the difference of cubes formula, and \(x^{2}-4\) can be factored into \((x-2)(x+2)\) using the difference of squares formula. The expressions, factorized, are: \[ [(x-2)(x^{2}+2x+4)] / [(x-2)(x+2)] * [(x+2)/(3x)] \]
02

Cancel like terms

Next, note that there are (x-2) terms in the numerator and the denominator which can be cancelled out, as can the (x+2) terms, leaving: \[ (x^{2}+2x+4) / (3x) \]
03

Simplify Expression

Having cancelled out the like terms, the final step is to simplify the remaining calculation. It cannot be simplified any further as it is already in its simplest form. Therefore, the final result is \( (x^{2}+2x+4) / (3x) \).

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