Prove that 9nnQ3nnQ Prove that 9nnQ3nnQSolutionTo show this

Solution

To show this, we need to show that { 9^n : n } { 3^n : n } and { 9^n : n } { 3^n : n }.

First Part: { 9^n : n } { 3^n : n }

Let 9^n { 9^n : n } for any n . Then, 9^n = 3^(2n). Since 2n as well (product of two rational numbers is rational!), 3^(2n) = 9^n { 3^n : n }.

Second Part: { 9^n : n } { 3^n : n }

Let 3^n { 3^n : n }, where n is again any rational number. Then, 3^n = (9^(1/2))^n = 9^(n/2). Since n , n/2 as well. So 3^n = 9^(n/2) { 9^n : n }.

Thus, { 9^n : n } = { 3^n : n }

Simple Explanation

if x = 9^n then x = 3^(2n)
if x = 3^n then x = 9^(n/2).
Here, for every rational number n, there exist another rational number n/2 and vice versa.
So 9^n = 3^(2n) = 3^n

Therefore, the both sets are equal