Determine the value of $a$ for which $5^{a}+5^{a+1}=\sqrt{4500}$.
Using exponent laws,

$$
\begin{aligned}
5^{a}+5^{a+1} & =\sqrt{4500} \\
5^{a}\left(1+5^{1}\right) & =30 \sqrt{5} \\
5^{a} \cdot 6 & =30 \sqrt{5} \\
5^{a} & =5 \sqrt{5} \\
5^{a} & =5^{1} \cdot 5^{1 / 2} \\
5^{a} & =5^{3 / 2}
\end{aligned}
$$

and so $a=\frac{3}{2}$.

ANSWER: $\frac{3}{2}$