Finding the square root of 729:

The square root of a number is a value that, when multiplied by itself, gives the original number. To find the square root of 729, we can use different methods, such as prime factorization, long division, or using a calculator. Here, we will use the method of prime factorization.

Prime factorization of 729:

To factorize 729, we can use the following steps:

1. Divide the number by the smallest prime number that divides it exactly. In this case, the smallest prime number is 3, so:

729 ÷ 3 = 243

2. Repeat step 1 with the quotient until we obtain a prime number. In this case:

243 ÷ 3 = 81
81 ÷ 3 = 27
27 is a prime number, so we stop here.

3. Write the prime factors in ascending order as a product. Here, we have:

729 = 3 × 3 × 3 × 3 × 3 × 3 = 3^6

Taking the square root of 729:

Now that we have the prime factorization of 729, we can use the property of exponents that states:

a^(m/n) = (a^m)^(1/n)

In other words, we can find the nth root of a number by taking the exponent m to the power of 1/n. Here, we want to find the square root of 729, which is the same as finding the 2nd root, so:

√729 = 729^(1/2) = (3^6)^(1/2) = 3^(6/2) = 3^3 = 27

Therefore, the square root of 729 is 27, which is option C.