Question

√2 is nt a rational number show that 2+√2 is nt a rational number
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Answer 1

Given: √2 is an irrational number.

To show: 2 + √2 is an irrational number.

Answer:

Let's assume 2 + √2 is a rational number.

$\sf 2\ +\ \sqrt{2}\ =\ \dfrac{a}{b},\ where\ 'a'\ and\ 'b'\ are\ co\ -\ prime\ integers\ and\ b\ is\ \ne\ 0.$

$\sf 2\ +\ \sqrt{2}\ =\ \dfrac{a}{b}\\\\\\2\ -\ \dfrac{a}{b}\ =\ \sqrt{2}\\\\\\\dfrac{2b\ -\ a}{b}\ =\ \sqrt{2}$

$\sf Since\ 'a'\ and\ 'b' \are\ inetegrs\ and\ \dfrac{2b\ -\ a}{b}\ is\ a\ rational\ number,\\\\\\\implies\ \sqrt{2}\ is\ also\ rational.\\\\\\It\ contradicts\ the\ fact\ that\ \sqrt{2}\ is\ irrational.\ [As\ per\ the\ question.]$

$\sf Therefore,\ our\ assumption\ is\ wrong.\\\\\\\bf 2\ +\ \sqrt{2}\ is\ an\ irrational\ number.$

Answer 2

Answer:

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Step-by-step explanation:

It is given that √2 is an irrational number.We will assume that '2+√2' is a rational.

it is expressed as

2+√2=a/b

√2=(a/b)-2

√2=(a-2b)/b

as a,b are integers

(a-2b)/b is a rational

√2 becomes rational

• This CONTRADICTS the fact that √2 is a irrational
• our assumption that 2+√2 is a rational is false
• hence we conclude that,2+√2 is a irrational.