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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

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

1 answer

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: \[ \boxed{x(x - 8)(x + 8)}. \]