Question

show that √5+√3 is an irrational number

Answer 1

let √5+√3 be a irrational number. √5+√4 =p/q. where p and q are costume integers.

√5+√3=p/q
squaring on both sides,
(√5+√3)^2=(p/q)^2
5+2√15+3=p^2/q^2
8+2√15=p^2/q^2
2√15=p^2-8q^2/q^2
√15=p^2-8q^2/2q^2

'we know that√15 is an irrational number'
*this contradiction has arisen or assumption is wrong √15 not equal to p^2-8q^2/2q^2.

[So,√15 is a irrational number]

Answer 2

Let's assume that√5+√3 are rational then there exist two positive integers a and b such that
√5+√3=a/b
(Square both sides)
(√3)²=(a/b-5)²
3=a²/b²+5-5√5a/b
2√5a/b=a²/b²+5-3
2√5a/b=a²/b²-2
2√5a/b=a²-b²/b²
√5=(a²-b²/b²)(b/2a)
√5=a²-b²/2ab
a²-b²/2ab is rational as a,b,2 are integers,
therefore √5 is irrational
So,√5+√3 is irrational
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