Question

prove that √3+2/5 is irrational​

Answer 1

Answer:

Given:3 + 2√5

To prove:3 + 2√5 is an irrational number.

Proof:

Letus assume that 3 + 2√5 is a rational number.

Soit can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.

so it contradictsour assumption.

Our assumption of3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved

Answer 2

Let take that $3 + 2 \sqrt{5}$

is rational number.

So, we can write this answer as,

$= > 3 + 2 \sqrt{5 } = \frac{a}{b}$

Here $a$&$b$two coprime number and $b \:  ≠  \: 0$

$= > 2 \sqrt{5} = \frac{a}{b} - 3$

$= > 2 \sqrt{5} = \frac{a - 3b}{b}$

$= > \sqrt{5} = \frac{a - 3b}{2b}$

Here $a$and $b$are integer so $\frac{a - 3b}{2b}$is a rational number but $\sqrt{5}$is a irrational number so it is contradict.

• Hence, $3 + 2 \sqrt{5}$is irrational.