Certainly! Let's perform the indicated operation and simplify the result step-by-step.
Given the expression:
[tex]$
\frac{\frac{9 x}{x^2-36}}{\frac{11 x}{x+6}}
$[/tex]
Step 1: Simplify the denominators and numerators
First, notice that \( x^2 - 36 \) can be factored. The difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \), applies here:
[tex]\[
x^2 - 36 = (x - 6)(x + 6)
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{9x}{(x - 6)(x + 6)}
\][/tex]
Step 2: Rewrite the division of fractions as a multiplication problem
Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, the original expression can be rewritten as:
[tex]\[
\frac{9x}{(x - 6)(x + 6)} \div \frac{11x}{x + 6} = \frac{9x}{(x - 6)(x + 6)} \times \frac{x + 6}{11x}
\][/tex]
Step 3: Simplify the product
When multiplying the fractions, the \( x + 6 \) terms cancel out and the \( x \) terms cancel each other:
[tex]\[
\frac{9x \cdot (x + 6)}{(x - 6)(x + 6) \cdot 11x } = \frac{9}{11(x - 6)}
\][/tex]
Step 4: Confirm the simplified form
After simplifying, we get the final result in factored form:
[tex]\[
\frac{9}{11(x - 6)}
\][/tex]
Therefore, the simplified answer is:
[tex]\[
\boxed{\frac{9}{11(x - 6)}}
\][/tex]
