[Algebra]
Problem: Let $T\neq 0$. Suppose that $a, b, c$, and $d$ are real numbers so that $\log _{a} c=\log _{b} d=T$. Compute

$$
\frac{\log _{\sqrt{a b}}(c d)^{3}}{\log _{a} c+\log _{b} d}
$$

Solution: Note that $a^{T}=c$ and $b^{T}=d$, thus $(a b)^{T}=c d$. Further note that $(a b)^{3 T}=(\sqrt{a b})^{6 T}=(c d)^{3}$, thus $\log _{\sqrt{a b}}(c d)^{3}=6 T$. Thus the given expression simplifies to $\frac{6 T}{2 T}=\mathbf{3}$ (as long as $T \neq 0$ ).

Final answer: 3