Expanding an expression of the form $(a \pm b)^{1/n}$

Question

Under what conditions can an expression like the one above (i.e. with a fractional exponent) be expanded? Apparently this kind of expansion is valid in some circumstances but I'm confused as to how given that $\sqrt{a^2+b^2}\neq a+b$.

Answer 1

Actually, you can expand $(a + b)^{z}$ for any $z \in \mathbb{C}$ using the Binomial Series: $$\sum_{k=0}^\infty {z \choose k} a^{z-k} \space b^{k}$$ With the note that $|a| > |b|$ and $a,b \in \mathbb{R}$

The Binomial Theorem