Problem 22 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The result of multiplying the given expressions is \( \frac{x^{2}+2x+4}{3x^{2}} \)

Step 1: Simplify the Fractions

The numerators and denominators simplify since \(x^{3}-8\) is a difference of cubes and \(x^{2}-4\) is a difference of squares. The functions simplify to: \((x^{3}-8) = (x-2)(x^{2}+2x+4)\) \(x^{2}-4 = (x-2)(x+2)\)Our equation thus becomes: \(\frac{(x-2)(x^{2}+2x+4)}{(x-2)(x+2)} \cdot \frac{x+2}{3x}\)

Step 2: Cancel Common Terms

In the first fraction, \(x-2\) is in both the numerator and the denominator and can be removed. In the both fractions \(x+2\) is accumulator and denominator and can be removed. The resultant fraction is:\(\frac{x^{2}+2x+4}{x} \cdot \frac{1}{3x}\)

Step 3: Further Simplification and Multiplication

When the fraction is simplified, it becomes: \(\frac{(x^{2}+2x+4)}{3x^{2}}\)