Question

Prove that
√12
are irrational
​

Answer 1

let root 12 rational
root 12 = a/b, where a and b are coprime and b is not equal to 0
√12=a/b
√12b= a
12b^2 = a^2
12 divides a^2
12 divide a
Again,
a = 12c
a^2= 144c^2
12b^2 = 144 c^2
b^2 = 12c^2
12 divides b^2
12 divide b

We know that a and b r coprime so they should not have any factor except 1 and itself. But they have factor as 12. This shows that √12 is rational. But this contradict the fact that √12 is irrational. This contradiction arises because of our wrong assumption that √12 is rational.
Hence √12 is irrational.

Hope this is helpful.
Plz mark me as brainliest.

Answer 2

Answer: Proof by contradiction

Step-by-step explanation: The above brainliest answer is wrong. The person said that if 12 divides a^2 then 12 divides a too. But that is not true, this only works with prime numbers.

For example, 12 divides 324(18^2) but 12 does not divide 18. So the proof is wrong. The real method to prove this is to simply write (12)^1/2 as 2*(3)1/2.
But we know that (3)^1/2 is irrational, so (12)^1/2 is irrational.