# Prove that 3?7? is irrational number.

$3\sqrt7\\*Let\;us\;assume\;3\sqrt7\;is\;rational.\\*So,\;we\;can\;write\;this\;number\;as\\*\Rightarrow 3\sqrt7=\frac{a}{b}..eq(1)\\*Here,\;a\;and\; b\;are\;two\;co-prime\;numbers\;and\;b\;is\;not\;equal\;to\;zero.\\*Simplify\; the\;..eq(1)\;divide\;by\;3\;both\;sides,\;we\;get\\*\Rightarrow \sqrt7= \frac{a}{3b}\\*Here,\;a\;and\;b\;are\;integers,\;so\;\frac{a}{3b}\;is\;a\; rational\;number,so\;\sqrt7\; should\;be \\* a\;rational\;number.\\*But\;\sqrt7\;is\;a\;irrational\;number,\;so\;it\;is\;contradictory.\\*Therefore,\;3\sqrt{7}\;is\;irrational\;number.\\*$

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