Question

What is the following quotient?

StartFraction 2 minus StartRoot 8 EndRoot Over 4 + StartRoot 12 EndRoot EndFraction
StartFraction StartRoot 3 EndRoot minus StartRoot 6 EndRoot Over 4 EndFraction
StartFraction 2 + StartRoot 3 EndRoot minus 2 StartRoot 2 EndRoot minus StartRoot 6 EndRoot Over 4 EndFraction
2 minus StartRoot 3 EndRoot minus 2 StartRoot 2 EndRoot + StartRoot 6 EndRoot
StartFraction negative 2 minus StartRoot 3 EndRoot + 2 StartRoot 2 EndRoot + StartRoot 6 EndRoot Over 2 EndFraction

Answer 1

(2 - √8)/(4 + √12) = (2 - 2√2)/(4 + 2√3) = (1 - √2)/(2 + √3).

Rationalizing the denominator: multiply top and bottom by (2 - √3):

(1 - √2)(2 - √3) = 2 - √3 - 2√2 + √6.

So the quotient equals 2 - √3 - 2√2 + √6 (equivalently (1 - √2)/(2 + √3)).