If you know only heights of $a$ and $b$, you may estimate heights of $a+b$, $a/b$ and $ab$. Assuming that $h$ is an absolute (Weil) height:
$$h(ab)\leq h(a)+h(b)$$
$$h(a/b)\leq h(a)+h(b)$$
$$h(a+b)\leq\log 2 +h(a)+h(b)$$
This bounds are sharp. You may find this, for example, in M. Waldschmidt "Diophantine approximation on linear algebraic groups", Chapter 3.