Question

Prove that root3 + root5 is irrational

Answer 1

Let _/3 + _/5 is a rational no.

•°• _/3 + _/5 = p/ q where p and q both are co- primes whose h.c.f. is 1.

Now ,

_/3 + _/5 = p / q

_/5 = p - _/3 by q

•°• Our assumption is wrong becoz irrational never = rational no

here _/5 and _/3 is an irrational no. but p is an rational no.

°•° _/3 + _/5 is an irrational no.

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Answer 2

Let us assume

$\sqrt{3 \:} \: + \sqrt{5} \: is \: \: a \: rational$

$\sqrt{3 \:} + \sqrt{5} = a \div b$

Squaring on both sides.

$3 + 5 + 2 \sqrt{15} = {(a \div b)}^{2}$

$7 + 2 \sqrt{15} = {a}^{2} \div {b}^{2}$

$\sqrt{15} = {a}^{2} - 7 {b}^{2} \div 2 {b}^{2}$

Irrational is not equal to rational

Which is contriduction.

Therefore,it is an irrational number.