Question

What is the simplest form of the radical expression?
show work please

Answer 1

let's recall that the conjugate of any expression is simply the same pair with a different sign between, so conjugate of "a + b" is just "a - b" and so on. That said, let's use the conjugate of the denominator

`\cfrac{√(2)+√(3)}{√(2)-√(3)}\cdot \cfrac{√(2)+√(3)}{√(2)+√(3)}\implies \cfrac{(√(2)+√(3))(√(2)+√(3))}{\underset{ \textit{difference of squares} }{(√(2)-√(3))(√(2)+√(3))}}\implies \cfrac{\stackrel{ F~O~I~L }{(√(2)+√(3))(√(2)+√(3))}}{(√(2))^2-(√(3))^2} \\\\\\ \cfrac{2+2√(2)\cdot √(3)+3}{2-3}\implies \cfrac{5+2√(6)}{-1}\implies \boxed{-5-2√(6)}`