A couple few things I've always wondered:

[WatermelonPig]

Just a couple tricks which are obviously incorrect, but why?

6/2 = 3

(-2)^3 = -8

(A^B)^C = A^(BC)

(-2)^3 = ((-2)^6)^(1/2) = (64)^(1/2) = 8

1/-1 = -1/1

sqrt(1/-1) = sqrt(-1/1)

1/i = i/1 And then multiply both sides by i

1 = -1

So what is happening here.

 

[Landau]

$$2=\sqrt{4}=2^{1/2}=((-2)^2)^{1/2}=-2^{2/2}=-2.$$

See here for an explanation.

 

[AdrianZ]

Just a couple tricks which are obviously incorrect, but why?

6/2 = 3

(-2)^3 = -8

(A^B)^C = A^(BC)

(-2)^3 = ((-2)^6)^(1/2) = (64)^(1/2) = 8

1/-1 = -1/1

sqrt(1/-1) = sqrt(-1/1)

1/i = i/1 And then multiply both sides by i

1 = -1

So what is happening here.

"Read Euler, Read Euler, he's master of us all".
Read Euler's elements of Algebra. he has explained all these in his treatise on Algebra.
the last one is incorrect because sqrt(A)sqrt(B) = sqrt(AB) is not valid for all complex numbers. (It is valid for real numbers though, but if we think of them as functions then these two functions are not equal because they can have different domains).