Question

`3 Prove that √3+2√5 is an ir rational number`​

Answer 1

$\large\leadsto\boxed{\rm\red{solution}}⇝$

Prove that 3 + 2√5 is irrational

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.

$\large\boxed{\rm\blue{3 + 2√5 \: is \: a \: irrational \: num}}$

Hence proved

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Answer 2

Step-by-step explanation:

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

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