Question

show that
$5 + \sqrt{2}$
is not a rational number. ​

Answer 1

$\huge{\underline{\underline{\red{♡Solution→}}}}$

$5 + \sqrt{2}$

Let (5 + √2) is a rational number.

And we know that A rational number can be written in the form of p/q where p and q are integers.

So ,

$5 + \sqrt{2} = \frac{p}{q} \\ \\ \sqrt{2} = \frac{p}{q} - 5 \\ \\ \sqrt{2} = \frac{p - 5q}{q}$

Therefore, p and q are Integers , (p-5q)/q is a rational number.

But the fact is that √2 is a irrational Number.

So our Suspposition us false.

Thus (5 + √2) is irrational Number.

$\rule{200}{1}$

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Answer 2

Step-by-step explanation:

Given:

• 5 + √2

To Show:

• It is not a rational number

Solution: Let 5 + √2 be a rational number.

• We know that a rational mumbe can be written in the form of p/q

$\small\implies{\sf }$ 5 + √2 = p/q

$\small\implies{\sf }$ √2 = p/q – 5

$\small\implies{\sf }$ √2 = (p – 5q)/q

Hence, p and q are integers then (p – 5q)/q is a rational number. Then √2 is also a rational number.

But this contradicts the fact that √2 is an irrational number.

Since, a contradiction arrises so our assumption is incorrect.

∴ 5 + √2 is an irrational number.

Proved ✓