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

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In summary, the radical symbol means to raise the number inside to the reciprocal of the little number of the radical. Therefore, \sqrt x = x^0.5 and \sqrt [0.5] x = x^2 are equivalent, as the reciprocal of 0.5 is 2. This can be understood by thinking of it as multiplying the number by itself, which results in x. However, there are also 8 distinct complex solutions to this problem.
  • #1
Someone502
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[tex] \sqrt x = x^ {.5} [/tex] and [tex] \sqrt [.5] x = x^2 [/tex]

They are the same but i want to know why.
 
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  • #2
Someone502 said:
[tex] \sqrt x = x^ {.5} [/tex] and [tex] \sqrt [.5] x = x^2 [/tex]

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.
 
  • #3
Someone502 said:
[tex] \sqrt x = x^ {.5} [/tex] and [tex] \sqrt [.5] x = x^2 [/tex]

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).
 
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  • #4
[tex]\sqrt x = x^5[/tex]
[tex](\sqrt x)^2 = (x^5)^2[/tex]
[tex]x = x^{10}[/tex] (notice at this point that x is either 0 or 1)
[tex](x)^{\frac{1}{5}} = (x^{10})^{\frac{1}{5}}[/tex]
[tex]\sqrt [5] x = x^2[/tex]
 
  • #5
quetzalcoatl9 said:
[tex]\sqrt x = x^5[/tex]
[tex](\sqrt x)^2 = (x^5)^2[/tex]
[tex]x = x^{10}[/tex] (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.
 
  • #6
For the life of me I can't remember ever seeing the notation [tex]\sqrt[n]{x} [/tex] where n was anything but a positive integer.
 
  • #7
quetzalcoatl9 said:
[tex]\sqrt x = x^5[/tex]
[tex](\sqrt x)^2 = (x^5)^2[/tex]
[tex]x = x^{10}[/tex] (notice at this point that x is either 0 or 1)
[tex](x)^{\frac{1}{5}} = (x^{10})^{\frac{1}{5}}[/tex]
[tex]\sqrt [5] x = x^2[/tex]

But the original poster wrote x^ 0.5 so how did you get to x = x^10 ?
 
  • #8
the original poster posted .5 not just 5. he wasn't implying
[tex] \sqrt x= x^5[/tex] he said [tex] \sqrt x= x^{.5} = x^{\frac{1}{2}} [/tex]

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.

[tex] \sqrt [2] x= x^{\frac {1}{2}} ; \sqrt [n] x= x^{1/n}[/tex]
the way you wrote the other equality is a bit odd, but its the same idea...
[tex] \sqrt [.5] x= \sqrt [\frac {1}{2}] x= x^2 [/tex]
 
  • #9
ok thanks even though it took me 10mins to understand it all
 
  • #10
Gale17 said:
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.
 

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

What is an exponent?

An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself.

What is the square root?

The square root of a number is the value that, when multiplied by itself, gives the original number. It is denoted by the symbol √ and is the inverse operation of squaring a number.

What does \sqrt x mean?

\sqrt x represents the square root of a number x. It is the value that, when squared, gives x.

What is the difference between \sqrt x and x^ {.5}?

The notation \sqrt x is the standard way of representing the square root, while x^ {.5} is the exponential notation for the square root.

What is \sqrt [.5] x?

\sqrt [.5] x represents the reciprocal square root of x. It is the value that, when squared, gives the reciprocal of x.

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