To find the square root of 180, you can follow these steps:

1. **Prime Factorization:** 
   - First, find the prime factors of 180.
   - \( 180 \div 2 = 90 \)
   - \( 90 \div 2 = 45 \)
   - \( 45 \div 3 = 15 \)
   - \( 15 \div 3 = 5 \)
   - \( 5 \div 5 = 1 \)
   - So, the prime factors of 180 are \( 2^2 \times 3^2 \times 5 \).

2. **Grouping the Factors:** 
   - We can group the prime factors into pairs.
   - \( \sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} \)
   - \( = \sqrt{(2 \times 3)^2 \times 5} \)
   - \( = \sqrt{36 \times 5} \)

3. **Simplifying the Square Root:**
   - \( \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} \)
   - \( = 6 \times \sqrt{5} \)

So, the simplified form of the square root of 180 is:

\[ \sqrt{180} = 6\sqrt{5} \]

If you want the decimal approximation:

\[ \sqrt{180} \approx 6 \times 2.236 = 13.416 \]

Would you like more details or have any questions?

Here are some additional questions you might have:

1. How do you find the prime factors of a larger number?
2. Can you explain why we group factors into pairs for square roots?
3. What is the decimal approximation of other square roots?
4. How do you simplify square roots involving fractions?
5. How does the square root function behave graphically?

**Tip:** When dealing with square roots, it's helpful to memorize the square roots of small perfect squares (e.g., \(\sqrt{1}\), \(\sqrt{4}\), \(\sqrt{9}\), etc.) to simplify more complex calculations.

Discover how to calculate the square root of 180 with a detailed step-by-step guide. This problem involves square roots and prime factorization concepts. Suitable for students in Grades 6-8.

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