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

Interpret the quotient as
[(2 − √8)/(4 + √12)] ÷ [(√3 − √6)/4].

Simplify radicals: √8 = 2√2, √12 = 2√3. Then
(2 − √8)/(4 + √12) = (2 − 2√2)/(4 + 2√3) = (1 − √2)/(2 + √3).

Also √3 − √6 = √3(1 − √2), so (√3 − √6)/4 = √3(1 − √2)/4.

Thus the quotient is
[(1 − √2)/(2 + √3)] · [4/(√3(1 − √2))] = 4/(√3(2 + √3)).

Rationalize:
4/(√3(2 + √3)) · (2 − √3)/(2 − √3) = 4(2 − √3)/√3
= (8√3 − 12)/3 = (4(2√3 − 3))/3.

So the quotient = (8√3 − 12)/3 (or 4(2√3 − 3)/3).