Question

Find the least common denominator.
StartFraction 4 Over x squared minus 36 EndFraction
and StartFraction 4 Over x left parenthesis x minus 6 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) for the fractions

\[
\frac{4}{x^2 - 36} \quad \text{and} \quad \frac{4}{x(x-6)},
\]

we need to factor both denominators.

1. The first denominator is \( x^2 - 36 \). This is a difference of squares and can be factored as follows:

\[
x^2 - 36 = (x - 6)(x + 6).
\]

2. The second denominator is \( x(x - 6) \). This denominator is already factored.

Now we have the two denominators factored:

- The first denominator: \( (x - 6)(x + 6) \)
- The second denominator: \( x(x - 6) \)

To find the LCD, we take each unique factor at the highest power it appears in any denominator.

The factors we have are:

- Factor \( (x - 6) \) from both,
- Factor \( (x + 6) \) from the first,
- Factor \( x \) from the second.

Combining these:

\[
\text{LCD} = x(x - 6)(x + 6).
\]

Therefore, the least common denominator is:

\[
\boxed{x(x - 6)(x + 6)}.
\]