The Conjugate Below

A bottom like 4 - √(6) is harder. Multiplying it by √(6) gives 4√(6) - 6, so the root only moves to the other term. You need the pair from the last chapter, the conjugate: the same two terms with the sign between them flipped. The conjugate of 4 - √(6) is 4 + √(6).

A sum times its difference loses its middle terms: (a + √(b))(a - √(b)) = a^2 - b. When a and b are whole numbers, so is that, and the root is gone. Take 10/(4 - √(6)) and multiply the top and the bottom by 4 + √(6):

10/(4 - √(6)) = (10(4 + √(6)))/(16 - 6) = (10(4 + √(6)))/10

The 10s cancel and the answer is 4 + √(6). Keep the top in its bracket until the bottom is worked out: the bottom is a whole number that often shares a factor with the number in front, as the 10s did here, and multiplying out early would hide that.