Question

3+2√5 prove that the rational and irrational number​

Answer 1

Answer:

Given:3 + 2√5

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

Proof:

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

Soit can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving3 + 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 But √5 is an irrational number.

so it contradictsour assumption.

Our assumption of3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved

hope this helps you

thank u

Answer 2

$\tt \huge \: Hola!$

• ProoF :

$\sf \large \: Let,$

$\mapsto \sf \: 3+2√5 (=x) \: \: is \: \: a \: \: rational \: \: number$

$\: \: \: \: \: \: \: \: \: \: \: \sf \: x = 3 + 2 \sqrt{5}$

$\implies \sf \: 3 + x = 2 \sqrt{5}$

$\implies \sf\frac{3 + x}{2} = \sqrt{5}$

$\sf hence , \: \: \sf \:rational \: \: number \: = irrational \: \: number$

→ But it's not possible

hence,

$\sf { \underline {\boxed{ \tt{ 3 - 2 \sqrt{5} \: \: is \: \: a \: \: irrational \: \: number}}}}$

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