Find the square root of 15876.

Answer

To find the square root of 15876, we can follow these steps:

### Step 1: Factor the number
We start by finding the prime factors of 15876. We can do this by dividing the number by the smallest prime numbers.

1. **Divide by 2**:
- \( 15876 \div 2 = 7938 \)

2. **Divide by 2 again**:
- \( 7938 \div 2 = 3969 \)

3. **Divide by 3**:
- \( 3969 \div 3 = 1323 \)

4. **Divide by 3 again**:
- \( 1323 \div 3 = 441 \)

5. **Divide by 3 again**:
- \( 441 \div 3 = 147 \)

6. **Divide by 3 again**:
- \( 147 \div 3 = 49 \)

7. **Divide by 7**:
- \( 49 \div 7 = 7 \)

8. **Divide by 7 again**:
- \( 7 \div 7 = 1 \)

So, the prime factorization of 15876 is:
\[ 15876 = 2^2 \times 3^4 \times 7^2 \]

### Step 2: Apply the square root
Now we can find the square root using the prime factorization:
\[ \sqrt{15876} = \sqrt{2^2 \times 3^4 \times 7^2} \]

### Step 3: Simplify the square root
Using the property of square roots, we can take the square root of each factor:
\[ \sqrt{15876} = \sqrt{2^2} \times \sqrt{3^4} \times \sqrt{7^2} \]
This simplifies to:
\[ = 2 \times 3^2 \times 7 \]

### Step 4: Calculate the final result
Now we calculate the values:
\[ 3^2 = 9 \]
Thus:
\[ \sqrt{15876} = 2 \times 9 \times 7 \]
Calculating this step by step:
1. \( 2 \times 9 = 18 \)
2. \( 18 \times 7 = 126 \)

So, the square root of 15876 is:
\[ \sqrt{15876} = 126 \]

### Final Answer:
The square root of 15876 is **126**.

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