Exponent Expression Help: Understanding 0^√0 = √0⁰ and Its Truth Value

[ritwik06]

Homework Statement

$$0^{\sqrt{0}}=\sqrt{0^{0}}$$

Is this expression true?

 

[CompuChip]

If you consider "undefined = undefined" to be true, then yes.

 

[Nick89]

That depends on your view... 0^0 can be defined as 1, or it can be undefined.

See http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerEither way, I think if you define 0^0 as undefined then the expression above does not hold... That would also mean 1/0 = 0^0 which I think doesn't really make sense... I'm not sure on this but I don't think you can say undefined = undefined!

 

[physixguru]

Take logarithm and simplify.

root 0 * log 0 = 0 *1= 0

Assumptions= using root 0= 0 on the basis that zero is a real number.
Taking log to the base 10.

R.H.S.> 1/2*0 * log 0
> 0*1
> 0

Assumptions same as above.

The points raised above by honourable members are very meaningful, this is one of the proof methods i learned at the IIT,delhi.

 

[HallsofIvy]

Take logarithm and simplify.

root 0 * log 0 = 0 *1= 0

?? log(0) is NOT 1. It is, one more time, "undefined".
(log(1)= 0, not the other way around.)

Assumptions= using root 0= 0 on the basis that zero is a real number.
Taking log to the base 10.

R.H.S.> 1/2*0 * log 0
> 0*1
> 0

Assumptions same as above.

The points raised above by honourable members are very meaningful, this is one of the proof methods i learned at the IIT,delhi.

 

[physixguru]

It was a bad typing error.