To simplify the given expression, we need to follow a series of steps methodically. The expression given is:
[tex]\[
\frac{\frac{3}{t+6}}{\frac{6}{t-6} - \frac{6}{t^2-36}}
\][/tex]
First, observe that [tex]\( t^2 - 36 \)[/tex] can be factored as [tex]\((t + 6)(t - 6)\)[/tex]:
[tex]\[
t^2 - 36 = (t + 6)(t - 6)
\][/tex]
This allows us to rewrite the expression inside the denominator:
[tex]\[
\frac{6}{t-6} - \frac{6}{t^2-36} = \frac{6}{t-6} - \frac{6}{(t+6)(t-6)}
\][/tex]
Next, find a common denominator for the fractions in the denominator:
[tex]\[
\frac{6}{t-6} - \frac{6}{(t+6)(t-6)}
\][/tex]
The common denominator is [tex]\((t + 6)(t - 6)\)[/tex]. Rewriting the fractions, we have:
[tex]\[
\frac{6(t + 6) - 6}{(t + 6)(t - 6)} = \frac{6t + 36 - 6}{(t + 6)(t - 6)} = \frac{6t + 30}{(t + 6)(t - 6)}
\][/tex]
Now, simplify the fraction in the denominator:
[tex]\[
\frac{6t + 30}{(t + 6)(t - 6)}
\][/tex]
We now substitute this back into the original expression:
[tex]\[
\frac{\frac{3}{t+6}}{\frac{6t + 30}{(t + 6)(t - 6)}}
\][/tex]
To simplify the complex fraction, multiply by the reciprocal of the denominator:
[tex]\[
\frac{3}{t+6} \cdot \frac{(t + 6)(t - 6)}{6t + 30}
\][/tex]
The [tex]\( t + 6 \)[/tex] terms in the numerator and denominator cancel out:
[tex]\[
\frac{3(t - 6)}{6t + 30}
\][/tex]
Finally, factor out a 6 in the denominator:
[tex]\[
6t + 30 = 6(t + 5)
\][/tex]
Thus, we get:
[tex]\[
\frac{3(t - 6)}{6(t + 5)} = \frac{t - 6}{2(t + 5)}
\][/tex]
So, the simplified expression is:
[tex]\[
\boxed{\frac{t - 6}{2(t + 5)}}
\][/tex]
