Question

Prove that three root and plus five root is irrational

Answer 1

√3+√5 is irrational
Let us assume it to be rational
So now it can be expressed in p/q form
√3+√5=p/q
√5=p/q-√3
Squaring on both sides
5=p²/q²-(2√3p/q)+3
5-3=p²/q²-(2√3p/q)
2-p²/q²=-(2√3p/q)
(p²-2q²)/q²=2p/q*√3
(p²-2q²)*q/(q²*2p)=√3
(p²-2q²)/(2pq)=√3
We know that p and q are integers so LHS is rational so RHS should also be rational
But √3 is irrational
So our assumption is wrong
√3+√5 is irrational

Answer 2

Hey there !

To prove :-
√3 + √5 is irrational.

Lets assume that √3 + √5 is rational.

Let ,
√3 + √5 = r , where r is a rational number.

Squaring both the sides ,

[√3 + √5 ]² = r²

3 + 2√15 + 5 = r²

8 + 2√15 = r²

2√15 = r² - 8

√15 = r² - 8 / 2

Here ,
RHS is purely rational .
But , LHS is irrational.
This is a contradiction.

Hence , our assumption was wrong.

Therefore , √3 + √5 is irrational.

Hope this Helps You !!!