Question

Prove that
$3 \: + \frac{1}{ \sqrt{2} } \:$
is an irrational number . ​

Answer 1

EXPLANATION

PROVE THAT 3 + 1/√2 IS IRRATIONAL

3 + 1/√2

3√2 + 1 /√2

x = 3√2 + 1 /√2

x^2 = ( 3√2 + 1 / √2 ) ^2

x^2 = (19 + 6√2 / 2 )

2x^2 = 19 + 6√2

2x^2 - 19 = 6√2

2x^2 - 19 / 6 = √2

since x is rational

therefore, x^2 is also rational

2x^2 - 19 / 6 = rational number

but √2 is irrational number

hence our assumptions is wrong

therefore,

3 + 1/√2 is irrational

Answer 2

PROVE THAT 3 + 1/√2 IS IRRATIONAL

3 + 1/√2

3√2 + 1 /√2

x = 3√2 + 1 /√2

x^2 = ( 3√2 + 1 / √2 ) ^2

x^2 = (19 + 6√2 / 2 )

2x^2 = 19 + 6√2

2x^2 - 19 = 6√2

2x^2 - 19 / 6 = √2

since x is rational

therefore, x^2 is also rational

2x^2 - 19 / 6 = rational number

but √2 is irrational number

hence our assumptions is wrong

therefore,

3 + 1/√2 is irrational