root3+root5whole square
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Here can solve this by using the formula (a+b)2 = a2+2ab+b2
Let (root 3 + root 5)2 be of the form a/b where a and b are co-primes and
b is not equal to zero.
(root 3 + root 5)2 = a/b
3 + 2 (root 3) (root 5) + 5= a/b
8 + 2 (root 3) (root 5) = a/b
2 (root 3) (root 5)= a/b-8
2 (root 3) (root 5)= 8b-a/b
2 (root 15) = 8b-a/b
root 15 = 8b-a/2b
Here the RHS is rational while the LHS is irrational.
This is not possible and therefore contradicts our assumption that (root 3 + root 5)2 is rational.
Therefore (root 3 + root 5)2 is irrational.
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