If n is a positive integer, then what is the value of (2/3)^n (1 1/2)^n-1? for positive integer n? Express your answer as a common fraciton?

Answer 1

(2/3)^n (1 1/2)^n-1
=(2/3)^n (3/2)^(n-1)
= (2/3)^n (2/3)^(1 - n)
= (2/3)(n + 1-n)
= (2/3)^1
= 2/3

Answer 2

btw, this is true for all values of n, not just positive integers.

e.g. let n = 2.3
(2/3)^2.3 (3/2)^(2.3-1)
= (2/3)^2.3 (3/2)^(1.3)
= (2/3)^2.3 (2/3)^-1.3
= (2/3)^(2.3 - 1.3)
= (2/3)^1 = 2/3