Question

Prove that (3+5√2) is an irrational number, given that √2 is an irrational number.​

Answer 1

Answer:

.

Step-by-step explanation:

Let us assume the contrary.

i.e; 5 + 3√2 is rational

$∴ 5 + 3√2 = \frac{a}{b}$

where ‘a’ and ‘b’ are coprime integers and b ≠ 0

$3√2 =  \frac{a}{b}  – 5$

$3√2 =  \frac{a - 5b}{b} </p><p>$

$√2 =  \frac{a - 5b}{3b} </p><p>$

$\scriptsize{Because \: ‘a’ and \: ‘b’ \: are \: integers  \frac{a - 5b}{3b}  is \: rational}$

That contradicts the fact that √2 is irrational.

The contradiction is because of the incorrect assumption that (5 + 3√2) is rational.

So, 5 + 3√2 is irrational.