How to expand an expression containing $x^2$ by binomial theorem

Question

I understand that the coefficient of, say $x^8$, in the expansion of $(1+x)^{10}$, would be ${10 \choose 8}$, but what about an expression like $(1+x^2)^{10}$? Would I have to square root the ${10 \choose 8}$? And if that is true, would I triple root it for expressions such as $(1+x^2)^{10}$ (if it is even possible)?

Answer 1

Let $x^2=y$. Then, by the Binomial theorem

$$(1+x^2)^{10}=(1+y)^{10}=\sum_{i=1}^{10}{10\choose i}y^i=\sum_{i=1}^{10}{10\choose i}(x^2)^i=\sum_{i=1}^{10}{10\choose i}x^{2i}$$ So, the coefficient of $x^8$ is ${10\choose 4}$