Question

What is the simplest form of the radical expression of `(sqrt2+sqrt5)/(sqrt2-sqrt5)`?

Answer 1

Multiply and divide by `sqrt(2)+sqrt(5)` to get:
`[sqrt(2)+sqrt(5)]^2/(2-5)=-1/3[2+2sqrt(10)+5]=-1/3[7+2sqrt(10)]`

Answer 2

Conjugate

Explanation:

Just to add on to the other answers,

We decided to multiply the top and the bottom by `sqrt(2)+sqrt(5)` because this is the conjugate of the denominator, `sqrt(2)-sqrt(5)`.

A conjugate is an expression in which the sign in the middle is reversed. If (A+B) is the denominator, then (A-B) would be the conjugate expression.

When simplifying square roots in the denominators, try multiplying the top and bottom by the conjugate. It will get rid of the square root, because `(A+B)(A-B) = A^2-B^2`, meaning you will be left with the numbers in the denominator squared.