Question

prove that √3+2√5 is an irrational number ​

Answer 1

Answer:

$\Huge\bf{\underline{{given}}}$

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

$\Huge\bf{\underline{{proof}}}$

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

So, it can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving 3 + 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 √5 is an irrational number.

So, it contradicts our assumption. Our assumption of 3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved.

Answer 2

Answer:

irrational number

Step-by-step explanation:

let us assume that 3+2√5 is a rational

3+2√5=p/q

here p and q are coprime numbers and q is not equal to zero

3+2√5=p/q

=>2√5=p/q-3

=>2√5=(p-3q)/q

=>√5=(p-3q)/2q

it shows (p-3q)/2q is a rational number but √5 is not a integer

this is a contradiction

this contradiction arises due to our assumption.