Question
What is the common denominator in the expression below? \[ \frac{4 x+8}{x^{2}-64}+\frac{x-5}{x+8} \] \( x+8 \) \( (x+8)(x-8) \) \( (x-8) \) There is no common denominator.
Ask by Vaughn Simpson. in the United States
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The common denominator is \( (x + 8)(x - 8) \).
Solution
First, factor the denominator in the first fraction:
x² - 64 = (x + 8)(x - 8).
Now, the first fraction is:
(4x + 8) / [(x + 8)(x - 8)]
and the second is:
(x - 5) / (x + 8).
To have a common denominator, both fractions must have (x + 8)(x - 8) as the denominator. The second fraction needs to be multiplied by (x - 8)/(x - 8).
So, the common denominator is (x + 8)(x - 8).
Therefore, the correct answer is:
(x + 8)(x - 8)
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Simplify this solution Beyond the Answer
To find the common denominator, first, let's factor the expression in the first fraction. The denominator \(x^2 - 64\) can be rewritten as \((x - 8)(x + 8)\) because it's a difference of squares. Therefore, the common denominator for both fractions in the expression is \((x + 8)(x - 8)\). Now, looking at both denominators, the second fraction is already \(x + 8\). To make them compatible, we can multiply the second fraction by \(\frac{x - 8}{x - 8}\) to get the same denominator for addition. This means we can combine both fractions seamlessly and simplify the expression from there!
