Find the square root of 1764.

Answer

To find the square root of 1764, we will follow these steps:

### Step 1: Factorization of 1764
We start by finding the prime factors of 1764. We can divide 1764 by the smallest prime number, which is 2.

- **1764 ÷ 2 = 882**
- **882 ÷ 2 = 441** (We can divide again by 2)
- **441 ÷ 3 = 147** (Next prime number is 3)
- **147 ÷ 3 = 49** (Again divide by 3)
- **49 ÷ 7 = 7** (Next prime number is 7)
- **7 ÷ 7 = 1** (Finally, divide by 7)

So, the prime factorization of 1764 is:
\[ 1764 = 2^2 \times 3^2 \times 7^2 \]

### Step 2: Taking the Square Root
Now we take the square root of the factorization:
\[ \sqrt{1764} = \sqrt{2^2 \times 3^2 \times 7^2} \]

Using the property of square roots that states \(\sqrt{a^2} = a\):
\[ \sqrt{1764} = \sqrt{(2 \times 3 \times 7)^2} \]

### Step 3: Simplifying
Now we can simplify this:
\[ \sqrt{1764} = 2 \times 3 \times 7 \]

### Step 4: Calculating the Product
Now we calculate the product:
- First, calculate \(2 \times 3 = 6\)
- Then, calculate \(6 \times 7 = 42\)

Thus, the square root of 1764 is:
\[ \sqrt{1764} = 42 \]

### Final Answer
The square root of 1764 is **42**.

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