Solving Exponent Question: \sqrt x = x^ {.5} and \sqrt [.5] x

[Someone502]

\sqrt x = x^ {.5} and \sqrt [.5] x = x^2

They are the same but i want to know why.

 

[jdavel]

\sqrt x = x^ {.5} and \sqrt [.5] x = x^2

They are the same but i want to know why.

Because that's what the radical symbol means: raise the number inside to the reciprocal of the little number of the radical.

 

[uart]

\sqrt x = x^ {.5} and \sqrt [.5] x = x^2

They are the same but i want to know why.

Think of it like this : (x^0.5) (x^0.5) = x^(0.5+0.5) = x^1 = x

So since (x^0.5) (x^0.5) = x then it follows that x^0.5 must be the square root of x (because when it's multiplied by itself it equals x).

 

[quetzalcoatl9]

\sqrt x = x^5
(\sqrt x)^2 = (x^5)^2
x = x^{10} (notice at this point that x is either 0 or 1)
(x)^{\frac{1}{5}} = (x^{10})^{\frac{1}{5}}
\sqrt [5] x = x^2

 

[dextercioby]

\sqrt x = x^5
(\sqrt x)^2 = (x^5)^2
x = x^{10} (notice at this point that x is either 0 or 1)

I doubt that.You left out 8 distinct complex (with nonzero imaginary part) solutions.

Daniel.

 

[shmoe]

For the life of me I can't remember ever seeing the notation \sqrt[n]{x} where n was anything but a positive integer.

 

[roger]

\sqrt x = x^5
(\sqrt x)^2 = (x^5)^2
x = x^{10} (notice at this point that x is either 0 or 1)
(x)^{\frac{1}{5}} = (x^{10})^{\frac{1}{5}}
\sqrt [5] x = x^2

But the original poster wrote x^ 0.5 so how did you get to x = x^10 ?

 

[Gale]

the original poster posted .5 not just 5. he wasn't implying
\sqrt x= x^5 he said \sqrt x= x^{.5} = x^{\frac{1}{2}}

this is true because as jdavel said, the radical symbol means: raise the number inside to the reciprocal of the little number of the radical.

\sqrt [2] x= x^{\frac {1}{2}} ; \sqrt [n] x= x^{1/n}
the way you wrote the other equality is a bit odd, but its the same idea...
\sqrt [.5] x= \sqrt [\frac {1}{2}] x= x^2

 

[Someone502]

ok thanks even though it took me 10mins to understand it all

 

[quetzalcoatl9]

the original poster posted .5 not just 5.

sorry, my screen resolution is such that it looked like a 5, not .5. my apologies for the additional confusion.

so essentially this "problem" boils down to knowing 0.5 = 1/2? well, duh, if i had realized that i wouldn't have bothered responding.