5^a = 1/25, 5^b = 25sqrt(5), 5^c = 1/sqrt(5). What is the value of a + b + c?

Two ways to do it. Manually work them all out: 5^a = 1/25 means a = -2, 5^b=25sqrt(5) means b = 2.5, 5^c = 1/sqrt(5) means c = -0.5. So a + b + c = -2 + 2.5 + 0.5 = 0.
Can alternatively remember that (x^t)(x^u) = x^t+u (when multiplying you add powers). This means that 5^a5^b5^c = 5^a+b+c = (1/25)(25sqrt(5))(1/sqrt(5)) = 1. We know that 5⁰ = 1, so a + b + c = 0.