Question

How do you rationalize `(2sqrt5)/ ( 2sqrt5 + 3sqrt2)`?

Answer 1

`(2sqrt(5))/(2sqrt(5)+3sqrt(2)) = 10-3sqrt(10)`

Explanation:

Note that the difference of squares identity tells us that:

`A^2-B^2=(A-B)(A+B)`

Hence we can rationalize the denominator of the given expression by multiplying both numerator and denominator by `2sqrt(5)-3sqrt(2)` ...

`(2sqrt(5))/(2sqrt(5)+3sqrt(2)) = (2sqrt(5)(2sqrt(5)-3sqrt(2)))/((2sqrt(5)-3sqrt(2))(2sqrt(5)+3sqrt(2)))`

`color(white)((2sqrt(5))/(2sqrt(5)+3sqrt(2))) = ((2sqrt(5))^2-(2sqrt(5))(3sqrt(2)))/((2sqrt(5))^2-(3sqrt(2))^2)`

`color(white)((2sqrt(5))/(2sqrt(5)+3sqrt(2))) = (20-6sqrt(10))/(20-18)`

`color(white)((2sqrt(5))/(2sqrt(5)+3sqrt(2))) = 10-3sqrt(10)`