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