Question

`(3 - sqrt(5))(2 - sqrt(5))` is ______.

Options

• rational

• irrational

• 1

• 11

Answer 1

`(3 - sqrt(5))(2 - sqrt(5))` is irrational.

Explanation:

Let’s re-evaluate carefully:

`(3 - sqrt(5))(2 - sqrt(5))`

Using the distributive property (FOIL):

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

= `6 - 3sqrt(5) - 2sqrt(5) + 5`

= `(6 + 5) - 5sqrt(5)`

= `11 - 5sqrt(5)`

 This expression contains the irrational term `sqrt(5)`, so it is irrational.