Evaluate `root3(729)`

Answer

To evaluate \( \sqrt[3]{729} \), we can follow these steps:

### Step 1: Identify the prime factorization of 729
First, we need to find the prime factors of 729. We can do this by dividing 729 by the smallest prime number, which is 3.

- \( 729 \div 3 = 243 \)
- \( 243 \div 3 = 81 \)
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

So, the prime factorization of 729 is:
\[ 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \]

### Step 2: Rewrite 729 in terms of its prime factors
Now we can express 729 as:
\[ 729 = 3^6 \]

### Step 3: Use the property of cube roots
The cube root of a number can be expressed in terms of its prime factorization. Thus:
\[ \sqrt[3]{729} = \sqrt[3]{3^6} \]

### Step 4: Apply the power of a power property
Using the property of exponents, we can simplify:
\[ \sqrt[3]{3^6} = 3^{6 \div 3} = 3^2 \]

### Step 5: Calculate the final result
Now we can calculate \( 3^2 \):
\[ 3^2 = 9 \]

Thus, the cube root of 729 is:
\[ \sqrt[3]{729} = 9 \]

### Final Answer
The value of \( \sqrt[3]{729} \) is \( 9 \).

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