Question

Prove that 3+2√5 is irrational number.​

Answer 1

$\large\blue{\textsf{✩ Your Answer ✓ }}$

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/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:

$\huge \bf \underline {\underline{Given :}}$

3 + 2√5

$\huge \bf \underline {\underline{to Prove :}}$

3 + 2√5 is an irrational number.

$\large \bf \underline {\underline{Proof:}}$

$\pink{\textsf{Let us assume that 3 + 2√5 is a rational number.}}$

$\blue{\textsf{So, it can be written in the form a/b}}$

$\green{\textsf{3 + 2√5 = a/b}}$

Here a and b are coprime numbers and b ≠ 0

Solving 3 + 2√5 = a/b we get,

$\large{⟹}$2√5 = a/b – 3

$\large{⟹}$2√5 = (a-3b)/b

$\large{⟹}$√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.

$\orange{\textsf{3 + 2√5 is an irrational number}}$

$\red{\textsf{Hence proved}}$

$\huge\mathtt\pink{\textsf{MissEleGant}}$