Have I solved the following correctly?
1. 5/a * 2/(a/b) = 10b
2. 5/a * 2/(a * b) = 10/(a^2 b)
3. 5/a * (2a)/b = 10b
Are equations 1. and 3. the same?
Your answers to problems 2 is correct, but problem 1 and problem 3 are incorrect. Many times, expressions can end up being equivalent even though they look wildly different. However, that is not the case here. Let's begin be rewriting the given expression from problem 1 to help better see what's going on:
\(\displaystyle \dfrac{5}{a} \cdot \dfrac{2}{\left( \dfrac{a}{b} \right)}\)
There's a fraction in the denominator of the second fraction, so we can use the fact that multiplication and division are inverses to help us out. When we multiply by something, we can divide by that same something to "undo" it and vice versa. But we also can see that multiplying by 1/x is the same as dividing by x, and dividing by 1/x is the same as multiplying by x. With any fraction \(\displaystyle \dfrac{p}{q}\), we know that \(\displaystyle \dfrac{1}{\left( \dfrac{p}{q} \right)} = \dfrac{q}{p}\). In other words, "one over" a fraction is the same as taking its reciprocal. Let's do that now and change it to multiplication:
\(\displaystyle \dfrac{5}{a} \cdot 2 \cdot \dfrac{b}{a}\)
Can you finish up from here? What do you get? Do you see why it's not the same as the given expression from problem 3?
You're also very close on problem 3, but I suspect you made an error when multiplying. The given expression looks like:
\(\displaystyle \dfrac{5}{a} \cdot \dfrac{2a}{b}\)
Multiplying straight across, as per the rules of multiplication, gives:
\(\displaystyle \dfrac{5 \cdot 2a}{a \cdot b}\)
What would your final step be? How does this new answer differ from your original answer? Do you see why this is correct and your original answer was not?