Find the cube root of `216 xx 1728`.

Answer

To find the cube root of \( 216 \times 1728 \), we will follow these steps:

### Step 1: Prime Factorization of 216
First, we need to find the prime factorization of \( 216 \).

1. Divide \( 216 \) by \( 2 \):
\[ 216 \div 2 = 108 \]
2. Divide \( 108 \) by \( 2 \):
\[ 108 \div 2 = 54 \]
3. Divide \( 54 \) by \( 2 \):
\[ 54 \div 2 = 27 \]
4. Divide \( 27 \) by \( 3 \):
\[ 27 \div 3 = 9 \]
5. Divide \( 9 \) by \( 3 \):
\[ 9 \div 3 = 3 \]
6. Divide \( 3 \) by \( 3 \):
\[ 3 \div 3 = 1 \]

So, the prime factorization of \( 216 \) is:
\[ 216 = 2^3 \times 3^3 \]

### Step 2: Prime Factorization of 1728
Next, we will find the prime factorization of \( 1728 \).

1. Divide \( 1728 \) by \( 2 \):
\[ 1728 \div 2 = 864 \]
2. Divide \( 864 \) by \( 2 \):
\[ 864 \div 2 = 432 \]
3. Divide \( 432 \) by \( 2 \):
\[ 432 \div 2 = 216 \]
4. Divide \( 216 \) by \( 2 \):
\[ 216 \div 2 = 108 \]
5. Divide \( 108 \) by \( 2 \):
\[ 108 \div 2 = 54 \]
6. Divide \( 54 \) by \( 2 \):
\[ 54 \div 2 = 27 \]
7. Divide \( 27 \) by \( 3 \):
\[ 27 \div 3 = 9 \]
8. Divide \( 9 \) by \( 3 \):
\[ 9 \div 3 = 3 \]
9. Divide \( 3 \) by \( 3 \):
\[ 3 \div 3 = 1 \]

So, the prime factorization of \( 1728 \) is:
\[ 1728 = 2^6 \times 3^3 \]

### Step 3: Combine the Prime Factorizations
Now, we can combine the prime factorizations of \( 216 \) and \( 1728 \):
\[ 216 \times 1728 = (2^3 \times 3^3) \times (2^6 \times 3^3) \]
Combining the powers of the same bases:
\[ = 2^{3+6} \times 3^{3+3} = 2^9 \times 3^6 \]

### Step 4: Find the Cube Root
Now, we need to find the cube root of \( 2^9 \times 3^6 \):
\[ \sqrt[3]{2^9 \times 3^6} = \sqrt[3]{2^9} \times \sqrt[3]{3^6} \]
Using the property of exponents:
\[ = 2^{9/3} \times 3^{6/3} = 2^3 \times 3^2 \]
Calculating the powers:
\[ = 8 \times 9 = 72 \]

### Final Answer
Thus, the cube root of \( 216 \times 1728 \) is \( 72 \).

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