Question

prove that â5 + â11 is irrational

Answer 1

       

Assume that √5 + √11 is rational.

So that √5 + √11 can be written as p / q, where p and q are coprime integers and q ≠ 0.

$\frac{p}{q}=\sqrt{5}+\sqrt{11} \\ \\ \\ (\frac{p}{q})^2=(\sqrt{5}+\sqrt{11})^2 \\ \\ \\ \frac{p^2}{q^2} = 16 + 2\sqrt{55} \\ \\ \\ \frac{p^2}{q^2}-16=2\sqrt{55} \\ \\ \\ \frac{p^2-16q^2}{q^2} =2\sqrt{55} \\ \\ \\ \frac{(p+4q)(p-4q)}{2q^2}=\sqrt{55}$

Here it contradicts the earlier assumption that √5 + √11 is rational, as LHS is rational while RHS is irrational.

∴ √5 + √11 is irrational.

Hence proved!!!

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