Question

1.prove that following are irrational?

$\sqrt[7]{5}$

Answer 1

$let \: \sqrt[7]{5} \: be \: rational$

$so,\: \sqrt[7]{5} \: = a/b$
$where \: a \: and \: b \: are \: co-prime \: integers \: and \: b≠0$
then,
$\sqrt[7]{5} \: =a/b$
$=> \sqrt{5} \: =a/7b$
$=> we \: know \: that \sqrt{5} \: is \: irrational \: and \: a/7b \: is rational \: (because, \: integer/integer \: = rational)$

$and$

rational ≠ irrational

$Therefore, \: Our \: supposition \: is \: wrong \: \sqrt[7]{5} \: is \: not \: rational. It \: is \: irrational.$

$Hope \: it \: helps \: uhh!!$

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