Question

show that (√3+5)² is irrational

Answer 1

Hey there!

We will prove that above number is irrational by contradiction method.

(√3+5)² = 3+25+10√3 = 28+10√3 .

Let us think that 28+10√3 is a rational number.
Then 28 + 10√3 = p/q ( p, q are co-primes,and integers )

10√3 = p/q -28
10√3 = p-28q/q
√3 = p-28q/10q .
Here p -28q/10q is rational so √3 is rational, but actually it is irrational. This contradiction has occurred because of our faulty assumption that (√3+5)² is rational .

Hence it is irrational.

( Why √3 is irrational? ).
Let √3 be rational
It can be expressed as √3 = a/b ( where a, b are integers and co-primes.
√3 = a/b
3 = a²/b²
3b² = a²
3 divides a²
so, 3 divides a .
a = 3k (for some integer)

a² = 9k²
3b² = 9k²
b² = 3k²

3 divides b²
3 divides b.

Now 3 divides both a & b this contradicts the fact that they are co primes.
this happened due to faulty assumption that √3 is rational. Hence, √3 is irrational.