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In the graph of \(y = a^x\) you can see at only one point the value of the y = 1. That is at x = 0. At no other value between - infinity and + infinity the value of y becomes 1. It's an always increasing curve. Irrational value also lie between - infinity and + infinity . Hence at no other irrational value is y = 1.
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The question is not can you rise up to iconic! The real question is will you ?

Hi Instructors, Request you to please clarify the below doubt on the usage of the property

\(x^y\) = 1

If X is not equal to 0 and\(x > 1\); Will the value of y be always 0. I mean, isnt it possible that y has some irrational value that makes \(x^y\) = 1

Example; \(2^(1/irrational value) = 1\). If not, what can we infer about the value of whole expression if \(x^(1/irrational value)\)

A number that can be expressed as a ratio of two numbers is rational. However, roots, \(\pi\), and endless non-repeating decimals are irrational numbers; they cannot be written in a ratio of two numbers. Moreover, an infinite amount of irrational numbers exist between integers. But, in no way, there is an infinite amount of irrational numbers that make integers, applying the logic that you used above.

I believe you are thinking as if rational and irrational numbers acted like odd and even numbers. Even though O + O = E or O + E = O, no such relationship exists between rational and irrational numbers.

I hope this helps.

gmatclubot

Re: Concept Clarity on X^y =1
[#permalink]
04 Jun 2013, 20:00

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