Find the cube-roots of :
`-2197`

Answer

To find the cube root of \(-2197\), we will follow these steps:

### Step 1: Identify the number
We need to find the cube root of \(-2197\).

### Step 2: Factor the positive part
First, we will find the prime factors of \(2197\).

### Step 3: Check divisibility
- \(2197\) is not divisible by \(2\) (since it's odd).
- Check for \(3\): The sum of the digits \(2 + 1 + 9 + 7 = 19\) is not divisible by \(3\).
- Check for \(5\): It does not end in \(0\) or \(5\), so it is not divisible by \(5\).
- Check for \(7\): \(2197 \div 7 \approx 314.14\) (not divisible).
- Check for \(11\): \(2197 \div 11 \approx 199.73\) (not divisible).
- Check for \(13\): \(2197 \div 13 = 169\) (divisible).

### Step 4: Factor further
Now we need to factor \(169\):
- \(169 = 13 \times 13\).

### Step 5: Write the complete factorization
So, we can express \(2197\) as:
\[ 2197 = 13 \times 13 \times 13 = 13^3 \]

### Step 6: Consider the negative sign
Since we are looking for the cube root of \(-2197\), we can write:
\[ -2197 = -1 \times 13^3 \]

### Step 7: Find the cube root
The cube root of \(-1\) is \(-1\), and the cube root of \(13^3\) is \(13\). Therefore:
\[ \sqrt[3]{-2197} = \sqrt[3]{-1 \times 13^3} = -1 \times 13 = -13 \]

### Final Answer
Thus, the cube root of \(-2197\) is:
\[ \sqrt[3]{-2197} = -13 \]

---