Question

Poe thse stale wnts G patul CNole: Notke tuut 6=2.3.use Uaique Paine Factorizt The oren, 4 arjue on expenuts of eur 2, or 3] arjue Use He conclusion fwm t N t3 13+ Sinrahral CNote: Suppo MHaumbun 9 whr's Ls Any COntaad ttesJ s inratengl" on th Contrun tis sone N-P. MGataral, e n prove Hat tonthe at by NG radmal. t's Mexct us dhamitsen ad prhe: wn tuwi namkus Cranem fashes oh tha 1. e r usgake beewse my Mkgn Can he eed hae CORN flr of prines is 1 if p9 ae comte Cnniu tncter and unk u hs blh to See if Au's a omtdahau IS ove L.3 ex 2ek 2ek CAnm beu pre fukr deUmpestn catahs 2

Answer 1

suppose, V6 is rational number

then by using de finition of rational number,

we can write

that there exists integers a and bwith no common divisors

such that V6

implies, V6b = a

taking square on both sides, we get

$6b^{2}=a^{2}$

hence, termon the right hand side should al so be even

That means a is even.

As, we knowthat square of any odd number is odd.

hence, we can say that a is even.

$let,a=2c,where\,c\,is\,an\,integer$

$As,a^{2}=6b^{2}$

$implies,(2c)^{2}=6b^{2}$

$implies,4c^{2}=6b^{2}$

$implies,2c^{2}=3b^{2}$

$That\,means\,b^{2}\,is\,even$

$hence,\sqrt{6}\,is\,a\,irrational\,number.$