Question

Find the least common denominator.
StartFraction 8 Over x squared minus 64 EndFraction
and StartFraction 9 Over x left parenthesis x minus 8 right parenthesis EndFraction

Question content area bottom
Part 1
The least common denominator is
  
enter your response here.
​(Type your answer in factored​ form.)

Answer 1

To find the least common denominator (LCD) of the fractions

\[
\frac{8}{x^2 - 64} \quad \text{and} \quad \frac{9}{x(x - 8)},
\]

we start by factoring the denominators.

1. **Factor \(x^2 - 64\)**: This expression is a difference of squares. It can be factored as follows:
\[
x^2 - 64 = (x - 8)(x + 8).
\]

2. **The second denominator** \(x(x - 8)\) is already in factored form.

Now we have:
- First fraction: \(\frac{8}{(x - 8)(x + 8)}\)
- Second fraction: \(\frac{9}{x(x - 8)}\)

Next, we identify all the factors present in the denominators:
- From the first fraction, we have \(x - 8\) and \(x + 8\).
- From the second fraction, we have \(x\) and \(x - 8\).

The unique factors to include in the LCD must be taken to their highest power:
- \(x\)
- \(x - 8\)
- \(x + 8\)

Therefore, the least common denominator (LCD) combining all these factors is:
\[
\text{LCD} = x(x - 8)(x + 8).
\]

So, the final answer is:
\[
\boxed{x(x - 8)(x + 8)}.
\]