# Whats wrong here?

1. Sep 28, 2004

### Anzas

whats wrong here?

2. Sep 28, 2004

### jcsd

$$(-4)^{\frac{2}{4}} \ne \sqrt[4]{(-4)^2}$$

Because both the rhs and the lhs are specifically the principle values.

3. Sep 28, 2004

### YourLooks

I think there is no definition of $$\sqrt{-4}$$ if complex analysis is not concerned right in the first place, so you have to stop right at that first place too, or your continuing proccess is then all meaningless, which is what the ugly guy in that picture is doing...:shy:

4. Sep 28, 2004

### BobG

The fourth root of 16 should be plus or minus two. :rofl:

If it weren't specifically defined that a had to be greater than 0 and not equal to one for $$f(x)=a^x$$ to be a valid function, you could have all kinds of possibilities:

$$-8^{\frac{1}{3}}$$ would be "75% chance of -2 and 25% chance of +2". A '1/3' exponent would give you -2, while a '2/6' exponent would give you plus or minus 2, etc.

5. Sep 28, 2004

### JasonRox

It depends on the convention, I guess.

6. Sep 28, 2004

### jcsd

We had this one before f(x) = x^(n/p) is specifically the principle branch of the function as is the radical symbol.

7. Sep 28, 2004

### Alkatran

Let's do it the other way then:

16^(1/4) = (16^(1/2))^(1/2) = (+-4)^(1/2) = 2 or -2 or 2i or -2i

8. Sep 29, 2004

### newPatrick

<1> Let x=-4 < 0, then let y1=sqrt(x) > 0 and y2=sqrt(x) < 0 (sqrt definition)and suppose these are correct. This means y^2=x. Paradox! y^2 < 0
<2> Rewrite the proof induction in the picture in this vars (x,y) form y=sqrt(x) = x^(2x(1/4))= (x^2)^(1/4)= ((-x)^2)^(1/4) = -x/2 (minus because x<0), this means y>0
If we suppose sqrt(negative) is allowed. From 2, y2 < 0 value is lost as Alkatran and many posts above indicated. But the truth is that sqrt(negative) isnot allowed, such assumption is impossible.
From1,2 say that picture gives a seemingly logical but incorrect induction

9. Sep 29, 2004

### matt grime

newPatrick, why are the complex numbers ordered? (in <1> you use < on complex quantities.)

10. Sep 29, 2004

### Alkatran

Here's the basic gist of what this guy did:

x = x
x^2 = x^2
sqr(x^2) = x or -x
x = -x

It's not a correct logic, or is NOT =

11. Sep 29, 2004

### HallsofIvy

Staff Emeritus
16, like any number, has 4 fourth roots: 2, -2, 2i and -2i.

In the real numbers we can specifically identify THE square or fourth root as the POSITIVE root. The complex numbers do NOT form an ordered field so we cannot distinguish complex roots in that way.