9. Prove that (√3+ √5)2 is an irrational
number
(CBSE 2016)
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Let us assume to the contrary that (√3+√5)² is a rational number,then there exists a and b co-prime integers such that,
(√3+√5)²=a/b
3+5+2√15=a/b
8+2√15=a/b
2√15=(a/b)-8
2√15=(a-8b)/b
√15=(a-8b)/2b
(a-8b)/2b is a rational number.
Then √15 is also a rational number
But as we know √15 is an irrational number.
This is a contradiction.
This contradiction has arisen as our assumption is wrong.
Hence (√3+√5)² is an irrational number.
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