Question

show that 5 + 2 root 2 is an irrational number where root 7 is given to irrational number

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Answer 1

EXPLANATIONS:-

$\rm{let : \: 5 + 2 \sqrt{7} = \frac{p}{q } } \\ \\ \therefore \: 5 + 2 \sqrt{7} = \frac{p}{q - 5} \\ \\ 2 \sqrt{7} = \frac{p - 5q}{q} \\ \\ \sqrt{7} = \frac{p - 5q}{2q}$

$\therefore \: \frac{p - 5q}{2q} \rm{ \ \: is \: a \: rational \: no.} \\ \\ \therefore \: \sqrt{7 } \sf{is \: also \: rational \: no.}$

$\sf{but \:it \: is \: not \: possible \: to \: have \: \sqrt{7} \: as \: irrational \: no.}$

$\therefore \rm{our \: assumption \: is \: wrong}$

$\therefore \: 5 + 2 \sqrt{7 \:} \rm{ \: is \: an \: irrational \: no.}$