Concept Clarity on X^y =1

Question

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)

MacFauz · Answer

x^a = y  means that when "x" is multiplied by itself "a" times, the answer is "y".

So as a corollary

y^{\frac{1}{a}} = x

Taking your example.,
You want to know whether  2^{\frac{1}{a}} = 1, where "a" is an irrational value.

Or in other words., you want to know whether  1^a = 2  for any "a".

The answer is NO. 1 raised to any power is 1.

AbhiJ · Answer

Basically you are saying 2 raised to some irrational value = 1. Any irrational should lie between -infinity and plus infinity.

see the graph of a^x

https://www.mathnotes.org/content/math/f ... _graph.gif

imhimanshu · Answer

Hi AbhiJ,

I didn't get your point over here. Could you please explain it in detail.

Thanks
imhimanshu

AbhiJ · Answer

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.

Bunuel · Answer

If  x^y = 1  and  x>1  then  y=0  only. For example,  2^y=1  -->  y=0 .

Hope it's clear.

KarishmaB · Answer

Responding to a pm:

You can also do another thing to understand this. Solve:

2^{1/x} = 1

Taking log on both sides

(1/x) log 2 = log 1 
 (.3/x) = 0

Is there a real value for x?

mejia401 · Answer

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.

Concept Clarity on X^y =1

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