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.