We are asked to solve for $c$ in terms of $a$ and $b$ given the following equation:

$\sqrt{a + \frac{b}{c}} = a \sqrt{\frac{b}{c}}.$

Step 1: Square both sides to eliminate the square roots.

$\left( \sqrt{a + \frac{b}{c}} \right)^2 = \left( a \sqrt{\frac{b}{c}} \right)^2$

This simplifies to:

$a + \frac{b}{c} = a^2 \frac{b}{c}.$

Step 2: Multiply both sides by $c$ to eliminate the fraction.

$c \left( a + \frac{b}{c} \right) = c \left( a^2 \frac{b}{c} \right)$

Simplifying:

$c a + b = a^2 b.$

Step 3: Isolate $c$.

$c a = a^2 b - b$

$c = \frac{b(a^2 - 1)}{a}.$

Thus, the solution for $c$ in terms of $a$ and $b$ is:

$c = \frac{b(a^2 - 1)}{a}.$

Would you like more details on any of the steps or further explanation?

Here are five related questions to expand on this concept:

1. How would the equation change if an additional term is added to the expression inside the square root?
2. What happens if $a = 1$? Does the equation have a valid solution for $c$?
3. Can we solve for $b$ in terms of $a$ and $c$ using the same method?
4. What other types of functions could require squaring both sides to solve?
5. How does the process change if higher powers or different roots are involved?

Tip: When solving equations involving square roots, always be cautious about extraneous solutions that may arise after squaring both sides.