Find the cube root of 389017.

Answer

To find the cube root of 389017, we will follow these steps:

### Step 1: Prime Factorization
We need to perform the prime factorization of 389017. This means we will divide the number by prime numbers until we can no longer divide.

1. Start with the smallest prime number, which is 2. Since 389017 is odd, it is not divisible by 2.
2. Next, try dividing by 3. The sum of the digits (3 + 8 + 9 + 0 + 1 + 7 = 28) is not divisible by 3, so 389017 is not divisible by 3.
3. Next, check for divisibility by 5. The last digit is not 0 or 5, so it is not divisible by 5.
4. Continue checking with prime numbers like 7, 11, 13, 17, and so on until we reach 73.

When we divide 389017 by 73, we find:
- 389017 ÷ 73 = 5329

Now, we need to factor 5329 further.

5. Continuing with 73, we find:
- 5329 ÷ 73 = 73

So, we can express 389017 as:
- 389017 = 73 × 73 × 73 = 73³

### Step 2: Finding the Cube Root
Now that we have the prime factorization, we can find the cube root.

The cube root of a number is the value that, when multiplied by itself three times, gives the original number.

Since we have:
- 389017 = 73³

The cube root of 389017 is:
- ∛(389017) = ∛(73³)

Using the property of cube roots, we can simplify this to:
- ∛(73³) = 73

### Final Answer
Thus, the cube root of 389017 is **73**.

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