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prove that √3+ √5 is irrational

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prove that √3+ √5 is irrational

Posted by Sekhar Borthakur 3 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Himanshu Chauhan 3 years, 4 months ago

Given √3 + √5 To prove: √3 + √5 is an irrational number. Let us assume that√3 + √5 is a rational number. So it can be written in the form a/b √3 + √5 = a/b Here a and b are coprime numbers and b ≠ 0 Solving √3 + √5 = a/b On squaring both sides we get, (√3 + √5)² = (a/b)² √3² + √5² + 2(√5)(√3) = a²/b² 3 + 5 + 2√15 = a²/b² 8 + 2√15 = a²/b² 2√15 = a²/b² – 8 √15 = (a²- 8b²)/2b a, b are integers then (a²-8b²)/2b is a rational number. Then √15 is also a rational number. But this contradicts the fact that √15 is an irrational number. Our assumption is incorrect √3 + √5 is an irrational number. Hence, proved.

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CBSE > Class 10 > Mathematics