My dilemma about $0^0$ We know that $0^0$ is indeterminate.
But if do this:
$$(1+x)^n=(0+(1+x))^n=C(n,0)\cdot ((0)^0)((1+x)^n) + \cdots$$
we get
$$(1+x)^n=(0^0)\cdot(1+x)^n$$
So, $0^0$ must be equal to $1$.
What is wrong in this?
Or am I mistaking $0^0$ as indeterminate?
Other threads are somewhat similar but not exactly the one I am asking.

The binomial theorem states that:
$$(a+b)^n=\sum_k\binom nka^kb^{n-k}$$
(assuming I made no typo). What you noticed is basically that, when $a=0$, this only works if $0^0=1$.
More specifically: When $k=0$, you're supposed to evaluate:
$$a^0b^n$$
when $a=0$. Now, while $0^0$ is an indeterminate form, it makes sense to assume that it's $1$ in this case, because $\displaystyle\lim_{a\to0}a^0=1$. In other words, while $0^0$ is indeterminate in general, in this case, it makes the most sense to take it as $1$.
At least, that's my understanding of it.
