Question

Is (3+√5)-(2+2√5) rational number or irrational number ​

Answer 1

Given:

• 3 + 2√5

To prove:

• 3 + 2√5 is an irrational number.

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

3 + √5 - 2 - √5 - √5

3-2 - √5

1 - √ 5

hence , it's irrational

$\mathsf{Hope \: it \: helps }$