Question

Prove is 3+5√2 is irrattional

Answer 1

Let 3 + 5√2 be a rational number.

Rational numbers are expressed in the form a/b,

where a and b are co-prime and b ≠ 0.

$\implies 3+5\sqrt{2}= \dfrac{a}{b}$

$\implies 5\sqrt{2}= \dfrac{a}{b}-3$

$\implies 5\sqrt{2}= \dfrac{a-3b}{b}$

$\implies \sqrt{2}= \dfrac{a-3b}{5b}$

RHS is a rational number

=> √2 is a rational number.

But this contradicts to the fact that √2 is an irrational number.

Hence, our assumption is wrong.

$\boxed{\boxed{\bold{Therefore, \ 3+5\sqrt{2} \ is \ an \ irrational \ number}}}}}}}$