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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 given expression simplifies to \[\frac{x}{3}+\frac{2}{3}+\frac{4}{3x}\].

Step by step solution

01

Factor the polynomials

First, factor the polynomials in the expression. The numerator \(x^{3}-8\) can be written as \((x-2)(x^{2}+2x+4)\) as these are the factors of \(x^{3}-8\). And the denominator \(x^{2}-4\) can be written as \((x-2)(x+2)\) because it is a difference of squares.
02

Substitute the factored polynomials

Substitute the factored forms back into the original expression: \[\frac{(x-2)(x^{2}+2x+4)}{(x-2)(x+2)} \cdot \frac{x+2}{3 x}\]. Remember that division by a fraction is equivalent to multiplication by its reciprocal.
03

Cancel out the common terms

Here, the terms \((x-2)\) and \((x+2)\) cancel each other, giving \[\frac{x^{2}+2x+4}{3x}\].
04

Simplify the resulting fraction

Simplify the expression further, this will give the final answer as \[\frac{x}{3}+\frac{2}{3}+\frac{4}{3x}\].

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Most popular questions from this chapter

Evaluate each expression without using a calculator. $$8^{\frac{2}{3}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Suppose a square garden has an area represented by \(9 x^{2}\) square feet. If one side is made 7 feet longer and the other side is made 2 feet shorter, then the trinomial that models the area of the larger garden is \(9 x^{2}+15 x-14\) square feet.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Evaluate each expression without using a calculator. $$27^{\frac{1}{3}}$$

Describe the kinds of numbers that have rational fifth roots.
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