Question

How do you simplify `7sqrt3-4sqrt6+sqrt48-sqrt54`?

Answer 1

`23sqrt(3)-7sqrt(6)`

Explanation:

Note:
`color(white)("XXX")color(green)(sqrt(6))=color(green)(sqrt(2)) * color(magenta)(sqrt(3))`
`color(white)("XXX")color(red)(sqrt(48))=color(red)(sqrt(16)) * color(magenta)(sqrt(3)) = color(red)(4) * color(magenta)(sqrt(3))`
`color(white)("XXX")color(blue)(sqrt(54))=color(blue)(sqrt(9) * sqrt(2)) * color(magenta)(sqrt(3))=color(blue)(3sqrt(2)) * color(magenta)(sqrt(3))`

Therefore:
`7color(magenta)(sqrt(3))-4color(green)(sqrt(6))+4color(red)(sqrt(48))-color(blue)(sqrt(54))`

`=color(magenta)(sqrt(3)) * ( 7 -4color(green)(sqrt(2))+4 *color(red)(4)-color(blue)(3sqrt(2)))`

`=color(magenta)(sqrt(3))(23-7sqrt(2))`

or
`=23sqrt(3)-7sqrt(6)`