Prove that (root 3+root 5) is an irrational num
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Let us assume that √3+√5 is rational.
Let √3+√5= a/b, where a,b are co-primes and b≠0.
Therefore, √3=(a/b)-√5.
Squarring on both sides, we get
3=(a²/b²)+5-2(a/b)√5
Rearranging,
(2a/b)√5=(a²/b²)+5-3
= (a²/b²)+2
√5=(a²+2b²)/2ab
Since, a,b are integers, (a²+2b²)/2ab is rational, and so √5 is rational.
This contradicts the fact that √5 is irrational. Hence,√3+√5 is irrational.
Let √3+√5= a/b, where a,b are co-primes and b≠0.
Therefore, √3=(a/b)-√5.
Squarring on both sides, we get
3=(a²/b²)+5-2(a/b)√5
Rearranging,
(2a/b)√5=(a²/b²)+5-3
= (a²/b²)+2
√5=(a²+2b²)/2ab
Since, a,b are integers, (a²+2b²)/2ab is rational, and so √5 is rational.
This contradicts the fact that √5 is irrational. Hence,√3+√5 is irrational.
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