# Intersection / Union of two regular expressions

Is there a way to directly form a new regular expression when given two regular expressions and the desired operation (Union or Intersection).

$$(a+b)^*aa(a+b)^*\cup (a+b)^*aa(a+b)^*$$

$$(a+b)^*aa(a+b)^*\cap (a+b)^*aa(a+b)^*$$

Without building NFAs and doing the operation through them?

Thank you.