Difference between unit-fractional exponent and root sign

[mishrashubham]

We were having our maths class a weeks ago and while studying quadratic equations our teacher asked us if there was an difference between x^{1/2} and \sqrt{x}.

Up until then I had always assumed them to be the same thing. However he said that there was a difference and said that he would tell us the answer later. But due to some reason he had to leave town. I have been wondering ever sine then but couldn't find anything.

Could anyone help? What is the difference?

Thank You.

 

[Mark44]

We were having our maths class a weeks ago and while studying quadratic equations our teacher asked us if there was an difference between x^{1/2} and \sqrt{x}.

In my view they are exactly the same. Let me stipulate that we're talking about real numbers x, with x >= 0. Each expression evaluates to a single, nonnegative number y such that y² = x.

Some people mistakenly believe that \sqrt{x} represents two numbers: one positive and one negative.

Up until then I had always assumed them to be the same thing. However he said that there was a difference and said that he would tell us the answer later. But due to some reason he had to leave town. I have been wondering ever sine then but couldn't find anything.

Could anyone help? What is the difference?

Thank You.

 

[micromass]

In my understanding, they are the exact same thing too. Note that there may be authors in advanced mathematics (mostly complex analysis) that will define things such that the two things are not equal. However, when doing quadratic equations and other real stuff, there is not difference...

 

[SteamKing]

I always find it suspicious when people have to leave town.

 

[Mark44]

I always find it suspicious when people have to leave town.

Especially just before they're about to tell us about the difference between x^1/2 and \sqrt{x}.

 

[micromass]

I got to use that excuse someday:

Hey, I've found an incredibly beautiful proof that there are not nontrivial integer solutions to x^n+y^n=z^n. But I have to leave town...

 

[Mute]

If x is real and positive, there's no difference. If x is negative or complex, there is a technical difference. The notation \sqrt{x} is a positive, singled-value real number. x^{1/2} does not carry this distinction. You have to define your choice of branch cut, and then you can evaluate the result. This, at least, is the convention that Churchill and Brown use, if I recall correctly. For example, the inverse hyperbolic functions are

\mbox{arcosh}(z) = \log(z + (z+1)^{1/2}(z-1)^{1/2})
where you cannot simplify (z+1)^{1/2}(z-1)^{1/2} = (z^2-1)^{1/2}, as this will give the wrong result for the chosen branch cut. (The wiki page for inverse hyperbolic functions uses the \sqrt{\.} notation, but I don't think that's standard.)

 

[mishrashubham]

I always find it suspicious when people have to leave town.

Especially just before they're about to tell us about the difference between x^1/2 and \sqrt{x}.

I got to use that excuse someday:

Hey, I've found an incredibly beautiful proof that there are not nontrivial integer solutions to x^n+y^n=z^n. But I have to leave town...

Haha, I was expecting that. Anyways, if you think its an excuse, you may happily assume so if it adds to your amusement.

Thank you.

 

[mishrashubham]

If x is real and positive, there's no difference. If x is negative or complex, there is a technical difference. The notation \sqrt{x} is a positive, singled-value real number. x^{1/2} does not carry this distinction. You have to define your choice of branch cut, and then you can evaluate the result. This, at least, is the convention that Churchill and Brown use, if I recall correctly. For example, the inverse hyperbolic functions are

\mbox{arcosh}(z) = \log(z + (z+1)^{1/2}(z-1)^{1/2})
where you cannot simplify (z+1)^{1/2}(z-1)^{1/2} = (z^2-1)^{1/2}, as this will give the wrong result for the chosen branch cut. (The wiki page for inverse hyperbolic functions uses the \sqrt{\.} notation, but I don't think that's standard.)

Thanks for the reply. We have not yet studied hyperbolic functions so I don't think I understood that bit. But I understood that as far as real numbers are concerned both are the same thing.
Thank you