# Basic Error? (1 Viewer)

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#### Aeneas

Could someone please sort out this contradiction which must come from some very basic error - but where and which error? If you raise -3 to the power of 1/2, this gives the square root of -3 which has no real value, but if you raise it to the power of 2/4, you are finding the fourth root of -3 squared, which is the fourth root of +9 which is real. What is wrong here?

Thanks in anticipation.

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#### HallsofIvy

Roughly speaking, the "laws of exponents" do not apply to complex numbers in the same way they apply to real numbers. But certainly look at the link Diffy mentioned.

#### Aeneas

Thanks for that, but I do no fully follow the replies in the link. One reply says that when you write p$$^{a/b}$$, a and b must be mutually prime. The demonstration that p$$^{a/b}$$= $$\sqrt{p^{a}}$$seems to work whether they are or not.

e.g. p$$^{a_{1}/b}$$ X p$$^{a_{2}/b}$$ ....X p$$^{a_{b}/b}$$ = p$$^{ab/b}= p^{a}$$.

Thus p$$^{a/b}$$ = $$\sqrt{p^{a}}$$. Where is the requirement there that they should be mutually prime? Or is it that the requirement is created by the need not to get into the contradiction?

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#### Diffy

Thanks for that, but I do no fully follow the replies in the link. One reply says that when you write p$$^{a/b}$$, a and b must be mutually prime.
I am not sure where that person was going with that reply. Certainly one can compute an answer for $$64^{2/6}$$.
I think that what he was getting at is that you are in tricky waters when you start using equalities. For example $$64^{2/6}$$ and $$64^{1/3}$$ aren't necessarily equal. Consider the polynomials that these two expressions are solutions to, Sqrt(x^6) = 64 and x^3 = 64. For the first wouldn't you say the answer is 4 or -4? and for the second there is only one answer 4. Therefore how could you say the two statements are equal?

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