Evaluate `root3((-512)/(343))`

Answer

To evaluate the expression \( \sqrt[3]{\frac{-512}{343}} \), we can follow these steps:

### Step 1: Break down the numbers
We know that:
- \( -512 = -8^3 \) (since \( 8 \times 8 \times 8 = 512 \))
- \( 343 = 7^3 \) (since \( 7 \times 7 \times 7 = 343 \))

### Step 2: Rewrite the expression
Now we can rewrite the expression using these values:
\[ \sqrt[3]{\frac{-512}{343}} = \sqrt[3]{\frac{-8^3}{7^3}} \]

### Step 3: Apply the cube root property
Using the property of cube roots, we can separate the numerator and denominator:
\[ \sqrt[3]{\frac{-8^3}{7^3}} = \frac{\sqrt[3]{-8^3}}{\sqrt[3]{7^3}} \]

### Step 4: Simplify the cube roots
Now, we can simplify each cube root:
\[ \sqrt[3]{-8^3} = -8 \quad \text{and} \quad \sqrt[3]{7^3} = 7 \]

### Step 5: Combine the results
Putting it all together, we have:
\[ \frac{-8}{7} \]

### Final Answer
Thus, the value of \( \sqrt[3]{\frac{-512}{343}} \) is:
\[ \frac{-8}{7} \]

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