prove that root2 + 3/root2 is irrational
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Answer:
To prove that √2 + 3/√2 is irrational.
Let us assume that √2 + 3/√2 is rational.
√2 + 3/√2 = a/b, where 'a' and 'b' are integers and 'b' ≠ 0.
Squaring both sides,
(√2 + 3)^2/(√2)^2 = (a/b)^2
(√2 + 3)^2/2 = a^2/b^2
2 + 9 + 6√2/2 = a^2/b^2
11 + 6√2 = 2a^2/b^2
6√2 = 2a^2/b^2 - 11
LCM for RHS: b^2
6√2 = 2a^2 - 11b^2/b^2
√2 = 2a^2 - 11b^2/6b^2
Since 'a' and 'b' are integers, 2a^2 - 11b^2/6b^2 is rational.
⇒ √2 is rational.
This contradicts the fact that √2 is irrational.
This contradiction has arisen due to our wrong assumption.
Therefore, √2 + 3/√2 is irrational.