Question

(√3-√2)^2
state whether the above numbers are rational or not​

Answer 1

(√3-√2)^2

3 -2×√3×√2 +2

5- 2√6

it is not rational

it's irrational

Answer 2

$( \sqrt{3} - \sqrt{2} ) {}^{2} = (\sqrt{3 {}}) {}^{2} + ( \sqrt{2} ) {}^{2} - 2( \sqrt{3} )( \sqrt{2} )$

$3 + 2 - 2 \sqrt{6} = 5 - 2 \sqrt{6}$

$let \: 5 - 2 \sqrt{6} \: is \: a \: rational \: no.$

$5 - 2 \sqrt{6} = \frac{p}{q}$

where p and q are co-prime

$- 2 \sqrt{6} = \frac{p}{q} - 5$

$- 2 \sqrt{6} = \frac{p - 5q}{q}$

$\sqrt{6} = \frac{p - 5q}{ - 2q}$

we know that root 6 is an irrational no.

$but \: \frac{p - 5q}{ - 2q} \: is \: a \: rational \: no.$

so our supposition is wrong

So

$( \sqrt{3} - \sqrt{2} ) {}^{2} is \: an \: irrational \: no.$

hope it helps

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