Question

How do you rationalize the denominator and simplify `1/(sqrt3-2)`?

Answer 1

Multiply the numerator and denominator by bye conjugate of the denominator to get:
`color(white)("XXX")-(sqrt(3)+2)`

Explanation:

The conjugate of a two-term expression `(a+b)` is `(a-b)` and visa versa.
The product of conjugates `(a+b)xx(a-b)` is `a^2-b^2`

For the given example, the conjugate of `(sqrt(3)-2)` is `(sqrt(3)+2)`

`1/(sqrt(3)-2)`
`color(white)("XXX")=1/(sqrt(3)-2)xx(sqrt(3)+2)/(sqrt(3)+2)`

`color(white)("XXX")=(sqrt(3)+2)/((sqrt(3)^2-2^2)`

`color(white)("XXX")=(sqrt(3)+2)/(3-4)`

`color(white)("XXX")=(sqrt(3)+2)/(-1)`

`color(white)("XXX")=-(sqrt(3)+2)`