Question

What is the following quotient? 2- sqrt 8 / 4 + sqrt 12

Answer 1

we have

`(2-√(8))/(4+√(12))`

Multiply the numerator and denominator by the conjugate of the denominator

so

`(2-√(8))/(4+√(12))*(4-√(12))/(4-√(12))=((2-√(8))(4-√(12)))/(4^(2)-(√(12))^(2))\\ \\= (2*4-2*√(12)-4*√(8)+√(12)*√(8))/(16-12)\\ \\=(8-4√(3)-8√(2)+4√(6))/(4) \\ \\=2-√(3)-2√(2)+√(6)`

therefore

the answer is

`2-√(3)-2√(2)+√(6)`

Answer 2

C is the answer.

We multiply the top and bottom of this by the conjugate (switch the sign of the square root):

(2-sqrt8)(4-sqrt12)/(4+sqrt12)(4-sqrt12)

Multiying the top we have
8-2sqrt12-4sqrt8+sqrt(8×12)

Multiplying the bottom we have:
16-4sqrt12+4sqrt12-12

On top, we simplify the radicals and get
8-4sqrt3-8sqrt2+4sqrt6

On bottom, we have
16-12=4

Now we divide everything on top by 4 and get
2-sqrt3-2sqrt2+sqrt6