Question

prove that root 3+2 root 5 is irrational

Answer 1

let 3 + 2 root 5 is rational
therefore 3 + 2 root 5=a/b(where a and b are any integer )
root 5=a-3/2b
be cos a-3/2b is rational
therefore root 5 will also be rational
but we know that,
root 5 is irrational
that is our supposition is wrong
that is 3+2 root 5 is irrational

Answer 2

Let us assume that 3+2√5 is rational.

Such that there exists two integers a and b such that 3+2√5 = a/b.

2√5 = a/b - 3

2√5 = a - 3b /b

√5 = a - 3b /2b

We know that √5 is irrational.
But it is equal to a-3b/2b where a and n are integers and so it is rational.

But this contradicts the fact that √5 is irrational.

Thus contradiction occurred because of our wrong assumption that 3+2√5 is rational.

HENCE 3+2√5 IS IRRATIONAL.

:)