Question

Examine whether the following numbers are rational or irrational (a) (√2+ 2) ² (b) 6/2√3

Answer 1

$\bigstar \frak{ \: \: \: \: \: {( \sqrt{2} + 2 )}^{2} } \\ \\ \leadsto \sf{\: {( \sqrt{2} )}^{2} + {(2)}^{2} + 2( \sqrt{2} )(2)} \\ \leadsto \sf \: \: \: 2 + 4 + 4 \sqrt{2} \\ \leadsto \sf \: \: \: 6 + 4 \sqrt{2} \: \: \: \: \: \: \: \: \boxed{\tiny( \frak{irrational})} \\ \\ . \degree. \: \: \bf \: \: {( \sqrt{2} + 2)}^{2} \: \: is \: \: \red{irrational}.$

And,

$\bigstar \: \: \: \: \frak{ \frac{6}{2 \sqrt{3} } } \\ \\ \leadsto \bf \: \frac{6}{2 \sqrt{3} } \times \frac{2 \sqrt{3} }{2 \sqrt{3} } \\ \\ \leadsto \bf \: \frac{6 \times 2 \sqrt{3} }{ {(2 \sqrt{3} )}^{2} } = \frac{ \cancel{12} \sqrt{3} }{ \cancel{12}} \\ \leadsto \bf \: \: \sqrt{3} \: \: \: \: \: \: \: \: \: \: \: \: \tiny \boxed{ \frak{irrational}} \\ \\ \\ \sf .\degree. \: \: \: \: \: \: \frac{6}{2 \sqrt{3} } \: \: \: is \: \: \: \red{irrational}.$