Suppose 4^a = 5, 5^b = 6, 6^c = 7, 7^d = 8, then the value of abcd is = ?

Solution (By Examveda Team)

$$\eqalign{ & 8 = {7^d} \cr & \,\,\,\,\,\, = {\left( {{6^c}} \right)^d} \cr & \,\,\,\,\,\, = {\left( {{5^b}} \right)^{cd}} \cr & \,\,\,\,\,\, = {5^{bcd}} \cr & \,\,\,\,\,\, = {\left( {{4^a}} \right)^{bcd}} \cr & \,\,\,\,\,\, = {4^{abcd}} \cr & \Rightarrow {4^{abcd}} = 8 \cr & \Rightarrow {\left( {{2^2}} \right)^{abcd}} = {2^3} \cr & \Rightarrow 2abcd = 3 \cr & \Rightarrow abcd = \frac{3}{2} \cr} $$