Find the square root of 1089.

Answer

To find the square root of 1089, we can use the method of prime factorization. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 1089
We start by dividing 1089 by the smallest prime number, which is 2. Since 1089 is odd, it is not divisible by 2. We then try the next smallest prime number, which is 3.

- **Divide by 3**:
- 1089 ÷ 3 = 363
- **Divide 363 by 3**:
- 363 ÷ 3 = 121
- **Divide 121 by 11**:
- 121 ÷ 11 = 11
- **Divide 11 by 11**:
- 11 ÷ 11 = 1

So, the prime factorization of 1089 is:
\[ 1089 = 3 \times 3 \times 11 \times 11 \]
or
\[ 1089 = 3^2 \times 11^2 \]

### Step 2: Applying the Square Root
Now we can find the square root of 1089 using its prime factorization:
\[ \sqrt{1089} = \sqrt{3^2 \times 11^2} \]

### Step 3: Simplifying the Square Root
Using the property of square roots, we can simplify:
\[ \sqrt{3^2} \times \sqrt{11^2} = 3 \times 11 \]

### Step 4: Calculate the Final Result
Now we multiply the results:
\[ 3 \times 11 = 33 \]

### Conclusion
Thus, the square root of 1089 is:
\[ \sqrt{1089} = 33 \]

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