Questions: Find the least common denominator of 9/(x+8) and 7/(x-8).

Find the least common denominator of 9/(x+8) and 7/(x-8).
Transcript text: Find the least common denominator of $\frac{9}{x+8}$ and $\frac{7}{x-8}$.
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Solution

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Solution Steps

To find the least common denominator (LCD) of two fractions, we need to determine the least common multiple (LCM) of their denominators. In this case, the denominators are \(x+8\) and \(x-8\). Since these are distinct linear expressions, their LCM is simply their product.

Step 1: Identify the Denominators

The denominators of the given fractions are \(d_1 = x + 8\) and \(d_2 = x - 8\).

Step 2: Calculate the Least Common Denominator

To find the least common denominator (LCD), we calculate the least common multiple (LCM) of the two denominators. Since both denominators are linear and distinct, the LCM is given by their product: \[ \text{LCD} = (x + 8)(x - 8) \]

Step 3: Expand the Expression

Now, we expand the product: \[ \text{LCD} = x^2 - 64 \]

Final Answer

The least common denominator of \(\frac{9}{x+8}\) and \(\frac{7}{x-8}\) is \(\boxed{x^2 - 64}\).

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