Question

prove that 3+2√5 is irrational​

Answer 1

Answer:

It is irrational

Step-by-step explanation:

3+4.472135....

7.472135...

Hence proved as the given number is non terminating and non recurring

Answer 2

PROOF BY CONTRADICTION METHOD :

Here, clearly the rational number is 3 and irrational number is 2√5 (already proved earlier that 2√5 is a irrational number).

We have to prove that, 3+2√5 is irrational.

If possible, let 3+2√5 be rational.

So, now, 3 as well as 3+2√5 are rational.

As, 3 and 3+2√5 both are rational, it follows that

As we know, difference of two rational numbers is rational, so,

3 - (3+2√5) = Rational

or, 3 - 3 - 2√5 = Rational

or, 0-2√5 = Rational

or, 2√5 = Rational

But, this contradicts that 2√5 is irrational (which is proved earlier). 2√5 can never be rational.

Hence, our supposition is wrong.

Therefore, 3+2√5 is irrational number and not rational.