show that √5-√7 is an irrational number
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Solution:
Let us assume (√5-√7) is a
rational number.
√5-√7 = a/b
Where a,b are integers and
b≠0
On Squaring both sides, we get
=> (√5-√7)² = (a/b)²
=> (√5)²+(√7)²-2*√5*√7=a²/b²
=> 5+7-2√35 = a²/b²
=> 12 - 2√35 = a²/b²
=> 12+a²/b² =2√35
=> (12b²+a²)/b² = 2√35
=> (12b²+a²)/2b² = √35
Since , a,b are rational number, (12b²+a²)/2b² is
rational. So, √35 is rational.
But , it contradicts that the fact that √35 is an irrational.
Therefore,
(√5-√7) is an irrational.
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