Why are powers of coprime ideals are coprime?

Let $R$ be your ring. I am assuming my ring is commutative. Saying $I,J$ are coprime ideals is the same as saying that there is no prime $q \subset R$ such that $I + J \subset q$. Well suppose there exists a prime $p \subset R$ such that $I^l + J^k \subset p$. Then $I^l \subset p$ and $J^k \subset p$. But this means $I \subset p$ and $J \subset p$ (why?), so $I + J \subset p$, contradicting the fact that $I,J$ are coprime.

You have $I + J = A$. Take each side to the $(k + \ell - 1)$-th power; then each term on the left is contained in either $I^k$ or $J^\ell$, and it follows that $I^k + J^\ell \supset A$. You could instead work with an expression $x + y = 1$, where $x \in I$ and $y \in J$, if you prefer.

If you are familar with ideal radicals then the proof is a one-liner:

$$\rm rad\ (I^m +\: \cdots\: + J^n) \ \supset\ I +\:\cdots\:+ J\: =\: 1\ \ \Rightarrow\ \ I^m +\:\cdots\: + J^n\: =\: 1 $$

Alternatively, and much more generally, it may be viewed as an immediate consequence of the Freshman's Dream binomial theorem $\rm\ (A + B)^n = A^n + B^n\ $ which is true more generally if $\rm\: A+B\:$ is invertible (e.g. $(1)$ or any principal ideal), e.g.

$\rm\qquad\qquad (A + B)^4 \ =\ A^4 + A^3 B + A^2 B^2 + AB^3 + B^4 $

$\rm\qquad\qquad\phantom{(A + B)^4 }\ =\ A^2\ (A^2 + AB + B^2) + (A^2 + AB + B^2)\ B^2 $

$\rm\qquad\qquad\phantom{(A + B)^4 }\ =\ (A^2 + B^2)\ \:(A + B)^2 $

So $\rm\qquad\ \ \ {(A + B)^2 }\ =\ \ A^2 + B^2\ $ if $\rm\ A+B\ $ is cancellative, e.g. if $\rm\ A+B = 1$

The same proof works generally since, as above

$\rm\qquad\qquad (A + B)^{2n}\ =\ A^n\ (A^n + \:\cdots\: + B^n) + (A^n +\:\cdots\: + B^n)\ B^n $

$\rm\qquad\qquad\phantom{(A + B)^{2n}}\ =\ (A^n + B^n)\ (A + B)^n $

For more see this prior answer.