Question

prove that √11 Is irrational; by using , it proves that 5+2√11 is also irrational
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Answer 1

To prove √11 is irrational_

Suppose,√11 is rational.

So,√11 = a×1/b×1 [Here / means upon]

a and b are co-prime.

They have a common factor '[1]'.

√11 = a/b

√11b = a

Therefore,

$\sqrt{11} {b}^{2} = {a}^{2} -(1)$

Therefore 7 is factor of a^2

So 11 Is a factor of a.

a=11c

11b^2-a^2

11b^2=(11c)^2

11b^2=121c^2

b^2=121/11c^2

Therefore b^2=11c^2

Therefore 11×c^2=b^2

So,11 is a factor of b^2.

Therefore 11 is a factor of b.

They have a common factor 11.

So, our assumption is wrong.

Hence proved that √11 is irrational.