`root(3)(175616)`

Answer

To find the cube root of \( 175616 \), we will follow these steps:

### Step 1: Prime Factorization
We start by finding the prime factors of \( 175616 \).

1. **Divide by 2** (the smallest prime number):
- \( 175616 \div 2 = 87808 \)
- \( 87808 \div 2 = 43904 \)
- \( 43904 \div 2 = 21952 \)
- \( 21952 \div 2 = 10976 \)
- \( 10976 \div 2 = 5488 \)
- \( 5488 \div 2 = 2744 \)
- \( 2744 \div 2 = 1372 \)
- \( 1372 \div 2 = 686 \)
- \( 686 \div 2 = 343 \)

2. **Now divide by 7** (next prime number):
- \( 343 \div 7 = 49 \)
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)

So, the prime factorization of \( 175616 \) is:
\[ 175616 = 2^9 \times 7^3 \]

### Step 2: Applying the Cube Root
Now, we can apply the cube root to the prime factorization:
\[ \sqrt[3]{175616} = \sqrt[3]{2^9 \times 7^3} \]

Using the property of cube roots:
\[ \sqrt[3]{a^m} = a^{m/3} \]
We can simplify:
\[ \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8 \]
\[ \sqrt[3]{7^3} = 7^{3/3} = 7^1 = 7 \]

### Step 3: Multiply the Results
Now, we multiply the results of the cube roots:
\[ \sqrt[3]{175616} = 8 \times 7 = 56 \]

### Final Answer
Thus, the cube root of \( 175616 \) is \( 56 \).

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