Question

Express the following with rational denominator:

`(3sqrt2 + 1)/(2sqrt5 - 3)`

Answer 1

We know that rationalization factor for `2sqrt5 - 3` is `2sqrt5 + 3`. We will multiply numerator and denominator of the given expression `(3sqrt2 + 1)/(2sqrt5 - 3)` by `2sqrt5 + 3` to get

`(3sqrt2 + 1)/(2sqrt5 - 3) xx (2sqrt5 + 3)/(2sqrt5 + 3) = (3sqrt2 xx 2sqrt5 + 3 xx 3sqrt2 + 2sqrt5 + 3)/((2sqrt5)^2 - (3)^2)`

`= (3 xx 2 xx sqrt2 xx sqrt5 + 3 xx 3sqrt2 + 2sqrt5 + 3)/(4 xx 5 - 9)`

`= (6sqrt(2 xx 5) + 9 sqrt2 + 2sqrt5 + 3)/(4 xx 5 - 9)`

`= (6sqrt10 + 9sqrt2 + 2sqrt5 + 3)/11`

Hence the given expression is simplified with rational denominator to `(6sqrt10 + 9sqrt2 + 2sqrt5 + 3)/11`