jmc

Question

Let x=4(5+1)( 45+1)( 85+1)( 165+1).Find (x+1)^48.

Accepted Answer

Multiplying the top and bottom by [16]5 - 1, we get a lot of simplification by difference-of-squares: aligned x& = 4([16]5-1)(5+1)([4]5+1)([8]5+1)([16]5+1)([16]5-1) &= 4([16]5-1)(5+1)([4]5+1)([8]5+1)([8]5-1) &= 4([16]5-1)(5+1)([4]5+1)([4]5-1) &= 4([16]5-1)(5+1)(5-1) &= 4([16]5-1)4 = [16]5 - 1. alignedThus, (x+1)^48 = ([16]5)^48 = 5^3 = 125.