Question

If 4^x-4^x-1=24, then find the value of (2x)^x. How to solve this problem in a quick and easy way for better understanding.

Answer 1

Given 4^x - 4^{x - 1} = 24
⇒ 4^x – (4^x/ 4¹) = 24 [Since a ^{m - n} = a^m / aⁿ]
⇒ 4^x [1 – (1/ 4)] = 24
⇒ 4^x [3/4)] = 24
⇒ 4^x = 24 × (4/3) = 32
⇒ (2²)^x = 2⁵
⇒ 2^2x = 2⁵
Comparing both the sides we get 2x = 5
Hence x = 5/2
Now consider, (2x)^x = [2(5/2)]^5/2
= [5)]^5/2
= [5⁵]^1/2
= √5⁵
= 25√5