Price that √3+√5 is irrational
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Assume that √3+√5 is rational.
Then √3+√5 = a/b; a and b are coprimes, b ≠ 0
√5 = a/b-√3
(√5)² = (a/b - √3)²
5 = a ²/b ²- 2√3a/b + 3
2 = a ²/b ²- 2 √3a/b
2 √3a/b = a ²/b ²-2
2 √3a/b = a ²-2b ²/b ²
2 √3 = a ²-2b ²×b /b ²×a
2 √3 = a ²-2b ² / ab ( b cancelled)
√3 = a ²-2b ² / 2ab
Here a ²-2b ² / 2ab is rational as a and b are integers.So √3 is also rational.But we konw that √3 is irrational. Therefore our assumption is wrong ie √3+√5 is irrational.
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