Determine if square root of, 12, end square root

Question

12
​
is rational or irrational and give a reason for your answer.
Answer
Attempt 2 out of 2

The number square root of, 12, end square root
12
​
is
rational
because
it is the square root of a non-perfect square

Answer 1

The square root of 12 is actually an irrational number.

Here's the reasoning:

1. A rational number can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

2. The number 12 can be factored into primes: \( 12 = 2^2 \times 3 \).

3. The square root of 12 can be simplified as follows: \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}. \]

4. The square root of 3 is known to be irrational. Therefore, \( 2\sqrt{3} \) is also irrational since multiplying a rational number (2) by an irrational number (\( \sqrt{3} \)) results in an irrational number.

So the correct conclusion is that \( \sqrt{12} \) is irrational because it can be expressed as \( 2\sqrt{3} \), where \( \sqrt{3} \) is irrational.