Irrational Number Raised To Irrational Power

Set 1.1
Question 60

See attachment.

I look for questions that create a discussion. This one certainly pushes one to think a bit more clearly. Work out (a) and (b).

NOTE: No need to work out every question posted tonight. You can take your time and spread your math work for several days if too busy.

 

60.
Can an irrational number raised to an irrational power yield an answer that is rational?

a.
let A=(sqrt(2))^(sqrt(2))

Let’s consider the number (sqrt(2))^(sqrt(2)) . This number is either rational or irrational. Let’s examine each case.

Case 1: rational or irrational
Recall that sqrt(2) is irrational. So if sqrt(2) ^(sqrt(2)) is rational, then we have proven that it’s possible to raise an irrational number to an irrational power and get a rational value. Done!

It turns out that it’s the second. The Gelfond–Schneider theorem tells us that for any two non-zero algebraic numbers a and b with a ≠ 1 and b irrational, the number a^b is irrational.
So sqrt(2) ^(sqrt(2)) is in fact irrational.

b.

Case 2: (sqrt(2))^(sqrt(2)) is irrational.

In this case, (sqrt(2))^(sqrt(2)) and sqrt(2) are both irrational numbers. So what if we raise the (sqrt(2))^(sqrt(2)) to the power of sqrt(2) ?

(sqrt(2))^(sqrt(2)) ^(sqrt(2)) ) = sqrt(2) ^ (sqrt(2)*sqrt (2)) = (sqrt( 2)) ^ 2 = 2 which is rational

 

Great reply. Can you check out my thread Using Triangle Inequality?