Question

root 2 + 21 is an irrational number or not​

Answer 1

Answer:

Step-by-step explanation:

Yes it is an irrational number.

Answer 2

$\sqrt{2} + 21 \: is \: irrational.$

$\\ \mathsf{\underline{Let's \: prove \: this} : } \\ Let \: us \: assume \: tote \: contrary \: that \: \sqrt{2} + 21 \: is \: rational. \\ So, \: \sqrt{2} + 21 = \frac{a}{b} \: \: \: \: \: \: | \: where \: a \: and \: b \: are \: coprime. \\ = > \: \: \: \: \: \: \: \: \: \: \sqrt{2} = \frac{a}{b} - 21 \\ = > \: \: \: \: \: \: \: \: \: \: \sqrt{2} = \frac{a - 21b}{b} \\ \\ \frac{a - 21b}{b} \: is \: rational \: so \: \sqrt{2} \: is \: rational. \\ But \: this \: contradicts \: the \: fact \: that \: \sqrt{2 } is \: irrational. \\ \: \: \: \: \: \: This \: contradiction \: has \: arisen \: due \: to \: our \: incorrect \: \\ assumption \: that \: \sqrt{2} + 21 \: is \: irrational. \\ \\ So, \: we \: conclude \: that \: \sqrt{2} + 21 \: is \: irrational. \\ \\ \mathsf{Hence \: proved.}$