Question

prove that root 3 plus root 5 is irrational

Answer 1

let √3+√5 be any rational number x
x=√3+√5
squaring both sides
x²=(√3+√5)²
x²=3+5+2√15
x²=8+2√15
x²-8=2√15
(x²-8)/2=√15
as x is a rational number so x²is also a rational number, 8 and 2 are rational nos. , so √15 must also be a rational number as quotient of two rational numbers is rational
but, √15 is an irrational number
so we arrive at a contradiction t
this shows that our supposition was wrong
so √3+√5 is not a rational number



OR OR OR OR OR OR OR OR OR OR OR U CAN DO IT LIKE THIS :
we know that, √3 and √5 are irrational numbers
so we know that sum of two irrational numbers is also irrational
√3+√5 is also irrational
hope this helps

Answer 2

Hi friend!!

Let √3+√5 be a rational number.
A rational number can be written in the form of p/q where p,q are integers.
√3+√5 = p/q
√3 = p/q-√5

Squaring on both sides,
(√3)² = (p/q-√5)²
3 = p²/q²+√5²-2(p/q)(√5)
√5×2p/q = p²/q²+5-3
√5 = (p²+2q²)/q² × q/2p
√5 = (p²+2q²)/2pq

p,q are integers then (p²+2q²)/2pq is a rational number.
Then √5 is also a rational number.
But this contradicts the fact that √5 is an irrational number.
So,our supposition is false.
Therefore, √3+√5 is an irrational number.