Question

Prove that of the numbers `5 + 3 sqrt (2)`  is irrational:

Answer 1

Let,`5 + 3 sqrt (2)` be rational.
Hence, 5 and `5 + 3 sqrt (2)` are rational.
∴ (`5 + 3 sqrt (2) – 5) = 3sqrt(2)` = rational       [∵Difference of two rational is rational]
∴ `1/3 × 3sqrt(2) = sqrt(2)` = rational     [∵Product of two rational is rational]
This contradicts the fact that `sqrt(2)` is irrational.
The contradiction arises by assuming `5 + 3 sqrt (2)` is rational.
Hence, `5 + 3 sqrt (2)` is irrational.