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Chapter 7: Problem 11

Find the least common denominator of the rational expressions. $$\frac{8}{y^{2}-9} \text { and } \frac{14}{y(y+3)}$$

Short Answer

Expert verified
The least common denominator of the rational expressions is \(y(y + 3)(y - 3)\).

Step by step solution

01

Factoring the Denominators

Firstly, we factor the denominators. \(y^2 - 9\) factors into \((y+3)(y-3)\) using the difference of squares method. The second denominator, \(y(y+3)\), is already factored into \(y\) and \(y+3\).
02

Find the LCM of the Factors

From factoring, we have the factors \(y\), \(y + 3\), and \(y - 3\). We then identify the least common multiple (LCM) of these factors. The LCM must include each factor the maximum number of times it appears in either factorization. Hence, the LCM of these factors (and thus, the LCD of the fractions) is \(y(y + 3)(y - 3)\).

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x+7}-1$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y+1)(2 y-1)}{(y-2)(y-3)}+\frac{(y+2)(y-1)}{(y-2)(y-3)}-\frac{(y+5)(2 y+1)}{(3-y)(2-y)}$$

One pipe can fill a swimming pool in 2 hours, a second can fill the pool in 3 hours, and a third pipe can fill the pool in 4 hours. How many minutes, to the nearest minute, would it take to fill the pool with all three pipes operating?

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Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)
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