How to simplify $x^a(u+v)^b$ and $(x+y)^a(u+v)^b$ when $a, b$ are binary digits?

Question

Say $a, b \in \{0,1\}$ and everything else is, say, a real number (though I don't think it should matter). Consider the following two expressions

$$ (\mathrm{i})\,\, x^a(u+v)^b \qquad \qquad (\mathrm{ii})\,\, (x+y)^a(u+v)^b $$

Is there a way to simplify any of these expressions?

For example, we can see \begin{align*} x^a(u+v)^b = \begin{cases} 1 & \text{if $ab = 00$}\\ u+v & \text{if $ab = 01$}\\ x & \text{if $ab = 10$}\\ xu + xv & \text{if $ab = 11$}. \end{cases} \end{align*}

I'd like to distribute the $x^a$ somehow. Same goes for (ii).

Answer 1

Hint: Find a linear function such that $f(0) = 1$ and $ f(1) = x$. Then, $f(a) = x^a$.