Find the cube - roots of :
`729/(-1331)`

Answer

To find the cube roots of \( \frac{729}{-1331} \), we will follow these steps:

### Step 1: Factorize the numbers
First, we need to factorize both the numerator and the denominator.

- **Numerator (729)**:
- \( 729 \div 3 = 243 \)
- \( 243 \div 3 = 81 \)
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

Thus, \( 729 = 3^6 \).

- **Denominator (-1331)**:
- \( 1331 \div 11 = 121 \)
- \( 121 \div 11 = 11 \)
- \( 11 \div 11 = 1 \)

Thus, \( 1331 = 11^3 \) and since it's negative, we have \( -1331 = -1 \times 11^3 \).

### Step 2: Rewrite the expression
Now we can rewrite the expression as:
\[ \frac{729}{-1331} = \frac{3^6}{-1 \times 11^3} \]

### Step 3: Apply the cube root
We need to find the cube root of the expression:
\[ \sqrt[3]{\frac{3^6}{-1 \times 11^3}} = \sqrt[3]{3^6} \div \sqrt[3]{-1} \div \sqrt[3]{11^3} \]

### Step 4: Simplify the cube roots
- The cube root of \( 3^6 \) is \( 3^{6/3} = 3^2 = 9 \).
- The cube root of \( -1 \) is \( -1 \) (since the cube root of a negative number remains negative).
- The cube root of \( 11^3 \) is \( 11^{3/3} = 11^1 = 11 \).

Putting it all together:
\[ \sqrt[3]{\frac{3^6}{-1 \times 11^3}} = \frac{9}{-11} = -\frac{9}{11} \]

### Final Answer
Thus, the cube root of \( \frac{729}{-1331} \) is:
\[ -\frac{9}{11} \]

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