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Chapter 7: Problem 12

Find the least common denominator of the rational expressions. $$\frac{14}{y^{2}-49} \text { and } \frac{12}{y(y-7)}$$

Short Answer

Expert verified
The least common denominator of the given rational expressions is \(y^{2}(y^{2} - 49)\).

Step by step solution

01

Factoring Denominators

Factoring the denominators of the given rational expressions, we get \(\frac{14}{(y - 7)(y + 7)}\) and \(\frac{12}{y(y - 7)}.\)
02

Determining the Least Common Denominator

The LCD is the least common multiple of the denominators. So, we look for the factors that appear in both expressions. They are \(y\), \(y - 7\), and \(y + 7\). The highest power of each factor in either denominator is the power of that factor in the LCD. Therefore the LCD of the given rational expressions is \(y(y - 7)(y + 7).\)
03

Simplifying the LCD

Finally, simplifying the LCD, we get \(y^{2}(y^{2} - 49)\).

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Most popular questions from this chapter

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y-3)(y+2)}{(y+1)(y-4)}-\frac{(y+2)(y+3)}{(y+1)(4-y)}-\frac{(y+5)(y-1)}{(y+1)(4-y)}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

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

Subtract: \(\frac{13}{15}-\frac{8}{45}\) (Section 1.2, Example 9)
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