a) Expand (2+√3)^5
b) Hence write (2-√3)^5 in the form a + b√3
For a) the answer is 362+209√3
For b) I guessed that it was 362-209√3 and I was right. The textbook solution just says this
(2-√3)^5 = (2+ (-√3))^5 = 362-209√3
If there was no simplification then it would make sense to me e.g.
(2+x)^5 = x^5 + 10 x^4 + 40 x^3 + 80 x^2 + 80 x + 32
therefore
(2-x)^5 = (-x)^5 + 10 (-x)^4 + 40 (-x)^3 + 80 (-x)^2 + 80 (-x) + 32
But the fact that this expansion has been simplified first confuses me in the fact that you can just replace the positive with the negative.
Like you can't do this replacement with a different surd e.g. this is clearly not right
(2-√5)^5 = 362-209√5
So does this only work if the surd changes sign? And is it something "obvious" that doesn't need to be proved because I don't immediately see why it's true.