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Chapter 7: Problem 10

Find the least common denominator of the rational expressions. $$\frac{3}{x-6} \text { and } \frac{4}{x^{2}-36}$$

Short Answer

Expert verified
The least common denominator of the two rational expressions is \((x+6)(x-6)\).

Step by step solution

01

Identify the Denominators

Identify the denominators of the given fractions. They are \(x-6\) and \(x^{2}-36\).
02

Factorize the Denominators

Factorize the denominators, if possible. The first one, \(x-6\), cannot be factorized further. However, the second denominator, \(x^{2}-36\), is a difference of two squares and can be factorized as \((x+6)(x-6)\).
03

Find the Least Common Denominator

The least common denominator (LCD) is the expression that includes all factors of each denominator, in the greatest quantity they occur. Combining the factors from both fractions, the LCD is \((x+6)(x-6)\).

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x}{x^{2}-y^{2}}+\frac{2 y}{y^{2}-x^{2}}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-1}{6}+\frac{x+2}{3}$$

Explain how to subtract rational expressions when denominators are the same. Give an example with your explanation.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6 x+5}{x-2}+\frac{4 x}{2-x}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used \(\frac{a}{d}=\frac{b}{e}\) to show that corresponding sides of similar triangles are proportional, but I could also use \(\frac{a}{b}=\frac{d}{e}\) or \(\frac{d}{a}=\frac{e}{b}\)
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