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Chapter 7: Problem 14

Find the least common denominator of the rational expressions. $$\frac{9}{y^{2}-25} \text { and } \frac{y}{y^{2}-10 y+25}$$

Short Answer

Expert verified
The least common denominator of \( \frac{9}{y^{2}-25} \) and \( \frac{y}{y^{2}-10 y+25} \) is \( (y-5)^2(y+5) \).

Step by step solution

01

Factorize the denominators

Factorize each of the denominators. The factorization of \( y^{2}-25 \) is \( (y-5)(y+5) \). The factorization of \( y^{2}-10y+25 \) is \( (y-5)^2 \).
02

Identify the common factors and additional factors

Looking at the factors, we can see that \( y-5 \) is a common factor. The other factors are \( y+5 \) from the first expression and another \( y-5 \) from the second expression.
03

Assemble the least common denominator

The least common denominator is found by taking the common factor \( y-5 \) and each of the additional factors \( y+5 \) and \( y-5 \). Hence, the LCD is \( (y-5)^2(y+5) \).

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Most popular questions from this chapter

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

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x}+\frac{3}{x-5}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{22 b+15}{12 b^{2}+52 b-9}+\frac{30 b-20}{12 b^{2}+52 b-9}-\frac{4-2 b}{12 b^{2}+52 b-9}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{x}+4$$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The difference between two rational expressions with the same denominator can always be simplified.
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