(Q7).Prove that the following are irrational.?

(iii) √3 + √5

Let us assume √3 + √5 is rational.

So we can write this number as.

√3 + √5 =
a
b
 ........1

Here, a and b are two co-prime numbers and b is not equal to .

Simplify the equation (1) squaring on both sides, we get

(√3 + √5)2 = (
a
b
 )2
(√3)2 + (√5)2 + 2 ×(√5)×(√3) =
a2
b2
+ + 2√15 =
a2
b2
+ 2√15 =
a2
b2
2√15 =
a2
b2
 -
√15 =
a2 - b2
2b
Here, a and b are integers so,
a2 - 8b2
2b
 is a number.

so √15 should be a number

But √15 is a number, so it is contradictor

Therefore, √3 + √5 is number.