Question

How many irrational terms have `(sqrt2+sqrt3)^15 =` ?

Answer 1

`2`

Explanation:

`(sqrt2+sqrt3)^2=5+2sqrt6`

so all the even powers have one irrational term of type `n sqrt6`

For an odd power we have

`(a+b sqrt6)(sqrt2+sqrt3) = (a+3b)sqrt2+(a+2b)sqrt3`

with two irrational terms of type `n sqrt2, m sqrt3` then

`(sqrt2+sqrt3)^15 = m sqrt2+n sqrt3` with two irrational terms.