Question

How do you simplify `(3sqrt3)/(-2+sqrt6)`?

Answer 1

See a solution process below:

Explanation:

The first step is to rationalize the denominator by multiplying the fraction by the appropriate form of `1`:

`(3sqrt(3))/(-2 + sqrt(6)) xx (-2 - sqrt(6))/(-2 - sqrt(6)) =>`

`(3sqrt(3)(-2 - sqrt(6)))/((-2)^2 + (-2 * -sqrt(6)) + (-2 * sqrt(6)) - (sqrt(6))^2) =>`

`((-2 * 3sqrt(3)) - (sqrt(6) * 3sqrt(3)))/(4 + 0 - 6) =>`

`(-6sqrt(3) - 3sqrt(6 * 3))/(-2) =>`

`(-6sqrt(3))/(-2) - (3sqrt(18))/(-2) =>`

`3sqrt(3) + (3sqrt(9 * 2))/2 =>`

`3sqrt(3) + (3sqrt(9)sqrt(2))/2 =>`

`3sqrt(3) + (3 * 3sqrt(2))/2 =>`

`3sqrt(3) + (9sqrt(2))/2`

Answer 2

`3sqrt3+(9sqrt2)/2`

Explanation:

`(3sqrt3)/(-2+sqrt6)`

`:.-(3sqrt3)/(-2+sqrt6) xx (-2-sqrt6)/(-2-sqrt6)`

`:.=(-6sqrt3-9sqrt2)/-2`

`:.=(cancel(-6)^3sqrt3)/cancel(-2)^1-(9sqrt2)/-2`

`:.=3sqrt3-(9sqrt2)/(-2)`

`:.=3sqrt3+((-9sqrt2)/1 xx 1/-2)`

`:.=3sqrt3+(9sqrt2)/2`

~~~~~~~~~~~~~~~~~~~~~~~~

`check`

`:.(3sqrt3)/(-2+sqrt6)=11.56011345`

`:.(-6sqrt3-9sqrt2)/-2=11.56011345`

`:.3sqrt3+(9sqrt2)/2=11.56011345`