Examine, whether the following numbers are rational or irrational: `(2-sqrt(2))(2+sqrt(2))` (ii) `(sqrt(3)+sqrt(2))^2`

Explanation

"`(i) (2 – sqrt2)(2 + sqrt2)`
`(2 – sqrt2)(2 + sqrt2) = 2^2 − sqrt2^2`
`[(a + b)(a – b) = a^2– b^2]`
`4 – 2 `
` 2` or `2/1`
Since, 2 is a rational number.
Hence, `(2 – sqrt2)(2 + sqrt2)` is a rational number.
`(ii) (sqrt(3)+sqrt(2))^2`
`(sqrt3)^2 + (sqrt2)^2+ 2sqrt3 xx sqrt2`
`[(a - b)^2 = a^2+ b^2- 2ab]`
` 3 + 2 + 2sqrt6`
`5 + 2sqrt6`
Since, the sum of a rational number and an irrational number is an irrational number.
Hence, `(sqrt3 + sqrt2)^2` is an irrational number."