
factoring
Hi all,
why can't a^2+b^2 be factored in fact non of the following factor
a^4+b^4
a^8+b^8
a^16+b^16
the factors are all even numbers and some even factors can be broken up to to produce a prime number factor ie 10 become 2^5 but i can't quite see why these sums can't be factored.
Thanks for the help.

Remember that factoring allows you to solve equations: $\displaystyle a^2 b^2= 0$ can be solved by $\displaystyle (a b)(a+ b)= 0$ so that a b= 0 or a= b and a+ b= 0 or a= b are solutions.
If it were possible to factor $\displaystyle a^2+ b^2$ with real coefficients (we can factor it as $\displaystyle a^2+ b^2= a^2 (b)^2= (a bi)(a+ bi)$) then it would be possible to find nonzero a and b so that $\displaystyle a^2+ b^2= 0$ and of course, that is not true. Since the square of any nonzero number is positive, if a and b are not both 0, then $\displaystyle a^2+ b^2$ must be positive, not 0.

thanks hallsofivy i also read something about there being no common factors can you help on this matter

I have no idea what you mean by that. Are you still referring to $\displaystyle a^2+ b^2$?
$\displaystyle a^2 b^2$ also has "no common factors" but can be factored.

Why can a^2b^2 be factored and a^2+b^2 can't I thought there needed to be a common factor(s) to factor something.

No, you do not have to have "common factors" at least not obvious ones.
IF you have a "common factor", like the "a" in ab+ ac, then you can simply use the "distributive law": ab+ ac= a(b+ c). But with binomials or trinomials it can be a bit more complicated.
While $\displaystyle a^2 b^2$ does NOT have an obvious "common factor", we could rewrite it as $\displaystyle a^2 ab+ ab b^2= (a^2 ab)+ (ab b^2$. Now the first binomial has a "common factor" of a and the second has a "common factor" of b: $\displaystyle a(a b)+ b(a b)$. And now we can see that the two terms have a "common factor" of ab: $\displaystyle (a+ b)(a b)$. But that is not always the simplest thing to do for something like $\displaystyle a^2 b^2$ it is simplest to remember the simple formula, $\displaystyle a^2 b^2= (a b)(a+ b)$.
Notice that if we try to do the same thing with $\displaystyle a^2+ b^2$ we would have $\displaystyle a^2 b^2= a^2 ab+ ab+ b^2= a(a b)+ b(a+ b)$ and now we cannot continue further.
But the real reason we cannot factor $\displaystyle a^2+ b^2$ or, more generally, $\displaystyle a^{2n}+ b^{2n}$ is what I said before: $\displaystyle a^{2n}+ b^{2n}= 0$ cannot have any real roots except a= b= 0. If we were able to factor, with real coefficients, we could find an infinite number of real number values for a and b that would satisfy the equation.