Question

How do you rationalize the denominator and simplify `sqrt(3)/(sqrt(7)-2)`?

Answer 1

`(sqrt21+2sqrt3)/3`

Explanation:

To rationalize the denominator, the best move is to multiply the numerator and denominator by the reciprocal of the denominator (which is the denominator but with the middle sign reversed).

`sqrt3/(sqrt7-2) * color(blue)((sqrt7+2)/(sqrt7+2))`

`(sqrt(3)(sqrt(7)+2))/((sqrt(7)-2)(sqrt(7)+2)`

From here, we can simplify the numerator and denominator.

`sqrt(alpha)sqrt(beta)=sqrt(alphabeta) -> sqrt3sqrt7=sqrt21`

`sqrt3(sqrt7+2)=color(red)(sqrt21+2sqrt3)`

`(sqrt7-2)(sqrt7+2) = 7+cancel(2sqrt7-2sqrt7)-4 = color(red)3`

`sqrt3/(sqrt7-2)=color(red)((sqrt21+2sqrt3)/3)`