prove that root 2 + root 7 is not rational number
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√2 + √7 = a/b (where a and b are co-prime)
√7 = a/b - √2
Squaring both sides
a²/b² - 2√2a/b + 2 = 7
a²/b² - 5 = 2√2a/b
(a² - 5b²) / 2ab = √2
√2 is an irrational number .
This contradicts the fact that √2 is irrational . So , our assumption is wrong.
Hence ,
√2 + √7 is irrational
√7 = a/b - √2
Squaring both sides
a²/b² - 2√2a/b + 2 = 7
a²/b² - 5 = 2√2a/b
(a² - 5b²) / 2ab = √2
√2 is an irrational number .
This contradicts the fact that √2 is irrational . So , our assumption is wrong.
Hence ,
√2 + √7 is irrational
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