Simplify : `root(3)(-2197)xxroot(3)(-125)+root(3)((27)/(512))`

Options

A. `(492)/(7)`
B. `(523)/(8)`
C. `(554)/(7)`
D. `(571)/(5)`

Answer

To simplify the expression \( \sqrt[3]{-2197} \times \sqrt[3]{-125} + \sqrt[3]{\frac{27}{512}} \), we will follow these steps:

### Step 1: Simplify each cube root
1. **Calculate \( \sqrt[3]{-2197} \)**:
- Since \( 2197 = 13^3 \), we have:
\[ \sqrt[3]{-2197} = -\sqrt[3]{2197} = -13 \]

2. **Calculate \( \sqrt[3]{-125} \)**:
- Since \( 125 = 5^3 \), we have:
\[ \sqrt[3]{-125} = -\sqrt[3]{125} = -5 \]

3. **Calculate \( \sqrt[3]{\frac{27}{512}} \)**:
- Since \( 27 = 3^3 \) and \( 512 = 8^3 \), we have:
\[ \sqrt[3]{\frac{27}{512}} = \frac{\sqrt[3]{27}}{\sqrt[3]{512}} = \frac{3}{8} \]

### Step 2: Substitute back into the expression
Now substituting these values back into the original expression:
\[ \sqrt[3]{-2197} \times \sqrt[3]{-125} + \sqrt[3]{\frac{27}{512}} = (-13) \times (-5) + \frac{3}{8} \]

### Step 3: Calculate the product
Calculate the product:
\[ (-13) \times (-5) = 65 \]

### Step 4: Combine the results
Now we combine the results:
\[ 65 + \frac{3}{8} \]

### Step 5: Convert 65 to a fraction
To add these, we convert 65 into a fraction with a denominator of 8:
\[ 65 = \frac{65 \times 8}{8} = \frac{520}{8} \]

### Step 6: Add the fractions
Now we can add:
\[ \frac{520}{8} + \frac{3}{8} = \frac{520 + 3}{8} = \frac{523}{8} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{523}{8} \]

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