Question

What is the simplest form of the expression? sqrt 2 - sqrt 6 / sqrt 2 + sqrt 6

Answer 1

`\displaystyle \\ ( √(2) -√(6))/(√(2) +√(6)) = ( (√(2) -√(6))(√(2) -√(6)))/((√(2) +√(6))(√(2) -√(6))) = \\ \\ \\ = ( (√(2) -√(6))^2)/((√(2))^2 -(√(6))^2) = ( 2-2√(2)√(6) + 6)/(2 -6) = \\ \\ \\ = ( 8-2√(12) )/(-4) = ( 8-2√(4* 3) )/(-4) = ( 8-4√(3) )/(-4) = \boxed{-2+√(3)}`

Answer 2

`-2+√(3)`

Explanation:

we have

`(√(2)-√(6))/(√(2)+√(6))`

Multiply the expression by the conjugate of denominator

`(√(2)-√(6))/(√(2)+√(6))*(√(2)-√(6))/(√(2)-√(6))\\ \\=-(√(2)-√(6))^(2)/4`

`=-(1/4)*(2-4√(3)+6)\\ \\=-(1/4)*(8-4√(3))\\ \\=-2+√(3)`