2005 AMC 10B Problem 17

Chaining the equations, $$\begin{aligned} 4^{abcd} &= \left(\left(\left(4^a\right)^b\right)^c\right)^d \\ &= \left(\left(5^b\right)^c\right)^d \\ &= \left(6^c\right)^d = 7^d = 8. \end{aligned}$$ Since $8 = 4^{\frac{3}{2}},$ we conclude $a \cdot b \cdot c \cdot d = \dfrac32.$ Thus, B is the correct answer.