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Chapter 0: Problem 15

Multiply or divide as indicated. $$\frac{x-2}{3 x+9} \cdot \frac{2 x+6}{2 x-4}$$

Short Answer

Expert verified
The result of multiplying \(\frac{(x-2)}{(3x+9)} \cdot \frac{(2x+6)}{(2x-4)}\) is \(\frac{1}{3}\).

Step by step solution

01

Simplify the Fractions

The given expression is \(\frac{(x-2)}{(3x+9)} \cdot \frac{(2x+6)}{(2x-4)}\). Now, let's simplify the fractions. It can be noticed that both denominators can be factored. The fraction \((3x+9)\) can be factored into \(3(x+3)\) and the fraction \((2x+6)\) can be factored into \(2(x+3)\). The original expression now becomes \(\frac{(x-2)}{3(x+3)} \cdot \frac{2(x+3)}{(2x-4)}\).
02

Cancel Common Factors

By the properties of fractions, we see that \((x+3)\) appears in both the numerator and the denominator of the expression, you can cancel out this term in both to further simplify the equation. The expression becomes \(\frac{x-2}{3} \cdot \frac{2}{2x-4}\).
03

Multiplying the simplified fraction

Multiplying these fractions by simply multiplying the numerators together and the denominators together, we get \((x-2)*2/((3)*(2x-4))\), which simplifies down to \(\frac{2(x-2)}{3(2x-4)} = \frac{2x-4}{6x-12}\).
04

Simplify

We can still simplify this fraction. The denominator \(6x-12\) can be factored into \(6(x-2)\) giving us \(\frac{2x-4}{6(x-2)}\). Both the numerator and the denominator share a common factor of \(x-2\) which can be cancelled out. After cancelling out \(x-2\) on top and bottom we are left with \(\frac{2}{6} = \frac{1}{3}\).

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Most popular questions from this chapter

Explain the quotient rule for exponents. Use \(\frac{5^{8}}{5^{2}}\) in your explanation.

Why must \(a\) and \(b\) represent non negative numbers when we write \(\sqrt{a} \cdot \sqrt{b}=\sqrt{a b ?}\) Is it necessary to use this restriction in the case of \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{a b} ?\) Explain.

Explain how to factor \(x^{3}+1\)

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right)$$

What is a perfect square trinomial and how is it factored?
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