Question
Prove that $(3+\sqrt{2})$ is irrational.
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Answer
If possible let $3+\sqrt{2}$ is rational number, and we take another rational number 3 for our calculation $\Rightarrow(3+\sqrt{2})-3=\sqrt{2}$ (difference of two rational number is a rational number)
$\therefore \sqrt{2}$ is rational
This contradicts the fact that $\sqrt{2}$ is irrational
Since the contradiction arises by assuming that $3+\sqrt{2}$ is rational.
Hence, $3+\sqrt{2}$ is irrational.
$\therefore \sqrt{2}$ is rational
This contradicts the fact that $\sqrt{2}$ is irrational
Since the contradiction arises by assuming that $3+\sqrt{2}$ is rational.
Hence, $3+\sqrt{2}$ is irrational.
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