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Since $$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$$ then $$x^5 - y^5 = (x^{5/3})^3 - (y^{5/3})^3= (x^{5/3} - y^{5/3})((x^{5/3})^2 + x^{5/3}y^{5/3}+(y^{5/3})^2)$$

It looks very wrong to me, but I can't find the algebra mistake

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up vote 6 down vote accepted

It's not wrong, and it's a nice thought! But...

Although this is not wrong, it's not a particularly nice factorization because the exponents are not natural numbers. You could just as correctly say that $x^3-y^3=(x^{3/2}-y^{3/2})(x^{3/2}+y^{3/2})$, but this similarly would not be very nice.

Now, it happens that $x^5-y^5$ does have a nice factorization in this way, which is why we can be picky on this point. Perhaps if there was no such factorization to be had, we could accept a solution with rational exponents.

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So what I have does, if I expand it, get $(x - y)(x^4 + yx^3 + y^2 x^2 + y^3 x + y^4$? – Hawk Jan 14 '13 at 5:55
Yes, this works. – Eric Stucky Jan 14 '13 at 5:56

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