`root(3)(12167)`

Answer

To find the cube root of 12167, we can use the prime factorization method. Let's go through the steps:

### Step 1: Prime Factorization
First, we need to factor 12167 into its prime factors. We can do this by dividing the number by prime numbers.

- Start with the smallest prime number, which is 2. Since 12167 is odd, we skip 2.
- Next, we try 3:
- 12167 ÷ 3 = 4055.67 (not divisible)
- Next, we try 5:
- 12167 ÷ 5 = 2433.4 (not divisible)
- Next, we try 7:
- 12167 ÷ 7 = 1738.14 (not divisible)
- Next, we try 11:
- 12167 ÷ 11 = 1106.09 (not divisible)
- Next, we try 13:
- 12167 ÷ 13 = 937.46 (not divisible)
- Next, we try 17:
- 12167 ÷ 17 = 716.29 (not divisible)
- Next, we try 19:
- 12167 ÷ 19 = 640.37 (not divisible)
- Next, we try 23:
- 12167 ÷ 23 = 529 (divisible)

Now we have:
12167 = 23 × 529

### Step 2: Factor 529
Next, we need to factor 529:
- 529 = 23 × 23 (since 23 × 23 = 529)

Putting it all together, we have:
12167 = 23 × 23 × 23 = 23³

### Step 3: Finding the Cube Root
Now that we have the prime factorization, we can find the cube root:
- The cube root of 12167 is the same as the cube root of 23³.

Thus:
\[ \sqrt[3]{12167} = 23 \]

### Final Answer
The cube root of 12167 is **23**.

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