Question

Prove that (root 5 -root 2) is an irrational number
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Answer 1

Assume that √5 - √2 is a rational number. Let it be x.

$\displaystyle x=\sqrt{5}-\sqrt{2} \\ \\ \\ x^2=(\sqrt{5}-\sqrt{2})^2 \\ \\ \\ x^2=5+2-2\sqrt{10}\\ \\ \\ x^2=7-2\sqrt{10} \\ \\ \\ 2\sqrt{10}=7-x^2 \\ \\ \\ \sqrt{10}=\frac{7-x^2}{2}$

At the last step, it seems that √10 can be written as a fraction. But it's absolutely wrong, isn't it?

Thus it creates a contradiction and thereby being our earlier assumption wrong.

Hence proved that √5 - √2 is an irrational number.