Prove that 2 + root 5 is irrational
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Let us assume that√2 + √5 is a rational number.
Then, there exist coprime positive integers a and b such that :-
√2 + √5 = a/b
a/b - √2 = √5
(a/b-√2)² = (√5)² ( squaring both sides )
a²/b² - 2a/b√2 + 2 = 5
a²/b²- 3 = 2a/b√2
a²-3b²/2ab = √2
√2 is a rational number.
[ a,b are integers ; a²-3b²/2ab is rational ]
This contradicts the fact that √2 is irrational number. Our assumption is wrong.
Hence, √2 + √5 is irrational !
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