Question

How do you rationalize the denominator and simplify `3/(t-sqrt2)`?

Answer 1

See a solution process below:

Explanation:

To rationalize the denominator, or, in other words, remove the radicals from the denominator, we need to multiply this fraction by the appropriate form of `1`.

We can use this quadratic rule to simplify this expression:

`(color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y)) = color(red)(x)^2 - color(blue)(y)^2`

`(color(red)(t) + color(blue)(sqrt(2)))/(color(red)(t) + color(blue)(sqrt(2))) xx 3/(color(red)(t) - color(blue)(sqrt(2))) => (3(color(red)(t) + color(blue)(sqrt(2))))/(color(red)(t)^2 - (color(blue)(sqrt(2)))^2) =>`

`(3t + 3sqrt(2))/(t^2 - 2)`