Question

given that √3 & √5 are irrational numbers , prove that √3+√5 is an irrational number

Answer 1

Answer:

Step-by-step explanation:

Let √3 + √5 be a rational number , say r

then √3 + √5 = r

On squaring both sides,  

(√3 + √5)2  = r2  

3 + 2 √15 + 5 =  r2  

8 + 2 √15  = r2    

2 √15 =  r2  - 8  

√15 = (r2 - 8) / 2  

Now (r2 - 8) / 2 is a rational number and √15 is an irrational number .  

Since a rational number cannot be equal to an irrational number . Our assumption that √3 + √5 is rational wrong .

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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