Find the cube-roots of :
`-512/(343)`

Answer

To find the cube root of \(-\frac{512}{343}\), we can follow these steps:

### Step 1: Identify the cube of the numerator and the denominator
We know that:
- \(512\) is the cube of \(8\) because \(8 \times 8 \times 8 = 512\).
- \(343\) is the cube of \(7\) because \(7 \times 7 \times 7 = 343\).

### Step 2: Rewrite the expression using cubes
We can express \(-\frac{512}{343}\) as:
\[ -\frac{512}{343} = -\frac{8^3}{7^3} \]

### Step 3: Simplify the expression
Using the property of fractions, we can rewrite the expression as:
\[ -\frac{8^3}{7^3} = -\left(\frac{8}{7}\right)^3 \]

### Step 4: Find the cube root
Now, we need to find the cube root of \(-\left(\frac{8}{7}\right)^3\):
\[ \sqrt[3]{-\left(\frac{8}{7}\right)^3} = -\frac{8}{7} \]

### Final Answer
Thus, the cube root of \(-\frac{512}{343}\) is:
\[ -\frac{8}{7} \]

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