Calculating the Square Root of 1089

To find the square root of 1089, we can use a method called prime factorization. Prime factorization involves breaking down a number into its prime factors and then finding the square root of each prime factor.

Step 1: Prime Factorization of 1089
To find the prime factorization of 1089, we can start by dividing it by the smallest prime number, which is 2. However, 1089 is an odd number, so it is not divisible by 2. We can move on to the next prime number, which is 3.

1089 ÷ 3 = 363

We can see that 1089 is divisible by 3. Now, we continue dividing by 3 until we reach a point where we can no longer divide evenly.

363 ÷ 3 = 121

121 is a perfect square, so we can stop here. Therefore, the prime factorization of 1089 is:

1089 = 3 × 3 × 121

Step 2: Finding the Square Root of Each Prime Factor
Now that we have the prime factorization of 1089, we can find the square root of each prime factor.

The square root of 3 is approximately 1.73 (rounded to two decimal places).

The square root of 121 is 11.

Step 3: Multiplying the Square Roots
To find the square root of 1089, we multiply the square roots of the prime factors.

1.73 × 11 = 19.03 (rounded to two decimal places)

Since 19.03 is not a whole number, we need to find the closest whole number to the square root of 1089. The closest whole number is 19.

Therefore, the square root of 1089 is 19.

Conclusion
The correct option is C) 33. The provided options are incorrect, as the square root of 1089 is not 29, 30, or 32. The correct answer is 33.