Question

prove that√3 + √5 is an irrational number.​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

Suppose that √3 + √5 is rational, say r.

Then :

√3 + √5 = r (note that r ≠ 0)

=> √5 = r - √3

=> (√5)² = (r - √3)²

=> 5 = r² + 3 - 2√3r

=> 2√3r = r² - 2

$= > \sqrt{3} = \frac{ {r}^{2} - 2 }{2r} .$

As r is rational and r ≠ 0.

So :

$\: \: \: \: \: \: \: \: \frac{ {r}^{2} - 2 }{2r} \: is \: rational$

$= > \sqrt{3} \: \: is \: rational.$

But this contradicts that √3 is irrational.

Hence :

Our supposition is wrong.

Therefore :

√3 + √5 is an irrational number.