The problem asks us to find the square root of 729, which is represented as $\sqrt{729}$.

1. Problem Description

The problem asks us to find the square root of 729, which is represented as $\sqrt{729}$.

2. Solution Steps

To find the square root of 729, we need to find a number that, when multiplied by itself, equals

7

2

9. We can use prime factorization to help us find this number.

First, we find the prime factorization of

7

2

9. 729 is divisible by 3 since the sum of its digits (7+2+9=18) is divisible by

3. $729 = 3 \times 243$

243 is also divisible by

3. $243 = 3 \times 81$

81 is $9 \times 9$, and 9 is $3 \times 3$.

$81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3$

Therefore, $729 = 3 \times 243 = 3 \times 3 \times 81 = 3 \times 3 \times 3 \times 27 = 3 \times 3 \times 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6$

Now we can rewrite the square root of 729 using the prime factorization:

$\sqrt{729} = \sqrt{3^6} = (3^6)^{\frac{1}{2}} = 3^{6 \times \frac{1}{2}} = 3^3 = 3 \times 3 \times 3 = 27$

Alternatively, we could notice that $729 = 9 \times 81 = 9 \times 9 \times 9 = 3^2 \times 3^2 \times 3^2 = (3 \times 3 \times 3)^2 = 27^2$

Therefore, $\sqrt{729} = \sqrt{27^2} = 27$.

3. Final Answer

27