Question

To prove that √3+√5 is irrational number

Answer 1

Hey !!!

let that √3+√5 is rational no.

and it is equal to a/b a and b is not equal to 0

:- √3+√5 = a/b

REARRANGING THE EQUATION

WE GET

therefore :- √3 = a/b - √5

SiNCE a and b are integer we get a/b -√5 is rational , and so √3 is rational

but this is contradicts the fact that √3 Is rational .

this contradiction has arises because of our incorrect assumption that √3 -√5 is rational.

so , we conclude that √3- √5 is irrational

Hope it helps !!!

#Rajukumar111€€€



NOW squaring in both side we got

Answer 2

___✨Here is your answer✨____


We have to prove that √3 + √5 is irrational.

1st Method :-
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→ Let us assume that √3 + √5 is rational number.

Now,

√3 + √5 = a/b

On squaring both sides we get,

3 + 5 + 2√15 = (a²/b²)

[As, (a + b)² = (a² + b² + 2ab)]

8 + 2√15 = (a²/b²)

2√15 = [(a² - 8b²) ÷ b²]

√15 = ½ [(a² - 8b²) ÷ b²]

Now, ½ [(a² - 8b²) ÷ b²] is a rational number

So, √15 is also a rational number.

But we know that √15 is irrational number.

So, our assumption is wrong √3 + √5 is a rational number.
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2nd Method :-
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→ Let us assume that √3 + √5 is a rational number.

So, 

√3 + √5 = a

On, squaring both sides we get,

8 + 2√15 = a²

2√15 = a² - 8

√15 = [(a² - 8) ÷ 2]

Now, [(a² - 8) ÷ 2] is rational number.

So, √15 is also a rational number.

But we know that √15 is a irrational number.

So, √15 is also a irrational number.

So, our assumption is wrong.

√3 + √5 is a irrational number.
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