Question

Prove that 3+2√5 is irrational.​

Answer 1

αɳŚωεɾ :

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

Answer 2

Your answer ⛄:

↬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/b3 + 2√5 = a/bHere a and b are coprime numbers and b ≠ 0.

$solving \: 3 + 2 \sqrt{5} \: = \frac{a}{b} \: we \: get \:$

$2 \sqrt{5} = \frac{a}{b} - 3 \\ 2 \sqrt{5} = \frac{a - 3b}{b} \\ \sqrt{5} = \frac{a - 3b}{2b} \\ \\ this \: shows \: that \: it \: is \: a\: rational \: no. \: but \: we \: know \: that \: \sqrt{5} \: is \: a \: irrational \: no.$

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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Hope it's helpful

@ Dhruvshi23 ⛄