Question

Find the square root of 56 using prime factorization.

Answer 1

To find the square root of 56 using prime factorization, we first need to factor 56 into its prime components.

1. **Factor 56:**
- We can divide by 2: \(56 \div 2 = 28\)
- Divide by 2 again: \(28 \div 2 = 14\)
- Divide by 2 again: \(14 \div 2 = 7\)
- Now, 7 is a prime number.

So, the prime factorization of 56 is:
\[
56 = 2^3 \times 7^1
\]

2. **Finding the square root:**
To find the square root, we can take half the exponent of each prime factor:
- For \(2^3\), half of 3 is \(1.5\), so we have \(2^{1.5} = 2^{1} \times \sqrt{2}\).
- For \(7^1\), half of 1 is \(0.5\), so we have \(7^{0.5} = \sqrt{7}\).

Thus, the expression for the square root of 56 is:
\[
\sqrt{56} = \sqrt{2^3 \times 7^1} = 2^{1.5} \times 7^{0.5} = 2 \sqrt{2} \times \sqrt{7} = 2 \sqrt{14}
\]

So, the square root of 56 can be simplified to:
\[
\sqrt{56} = 2\sqrt{14}
\]

In summary, the square root of 56 is:
\[
\sqrt{56} = 2\sqrt{14}
\]

Answer 2

Wrong, re-do it

Answer 3

Let's redo the calculation for the square root of 56 using prime factorization step by step.

### Step 1: Prime Factorization of 56
First, we need to factor 56 into its prime components.

1. Start with 56.
- \(56 \div 2 = 28\)
2. Factor 28:
- \(28 \div 2 = 14\)
3. Factor 14:
- \(14 \div 2 = 7\)

Now, 7 is a prime number.

So, the prime factorization of 56 is:
\[
56 = 2^3 \times 7^1
\]

### Step 2: Finding the Square Root
To find the square root of 56, we can take the square root of each prime factor raised to its respective exponent.

1. For \(2^3\):
- The square root is \(2^{3/2} = 2^{1.5} = 2^{1} \cdot 2^{0.5} = 2 \sqrt{2}\).

2. For \(7^1\):
- The square root is \(7^{1/2} = \sqrt{7}\).

Now, combining these results gives us:
\[
\sqrt{56} = \sqrt{2^3 \times 7^1} = 2^{3/2} \times 7^{1/2} = 2 \sqrt{2} \times \sqrt{7}.
\]

### Step 3: Final Simplified Form
Thus, we have:
\[
\sqrt{56} = 2 \sqrt{14},
\]
where \(\sqrt{14} = \sqrt{2 \times 7} = \sqrt{2} \cdot \sqrt{7}\).

So, the square root of 56 is:
\[
\sqrt{56} = 2 \sqrt{14}.
\]

### Conclusion
The steps confirm that the square root of 56, written in terms of its simplest radical form, is:
\[
\sqrt{56} = 2\sqrt{14}.
\]

If you're looking for a numerical approximation:
\[
\sqrt{56} \approx 7.483.
\]

Please let me know if you need further assistance!