Question
Determine if square root of, 12, end square root
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
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
Answers
GPT-4o mini
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.
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.
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