The value of `(root(3)(81) + root(3)( - 192) + root(3)( 375) )/( root(3) (24) )` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the expression \((\sqrt[3]{81} + \sqrt[3]{-192} + \sqrt[3]{375}) / (\sqrt[3]{24})\), we will break it down step by step.

### Step 1: Simplify each cube root
1. **Calculate \(\sqrt[3]{81}\)**:
- \(81 = 3^4\)
- Therefore, \(\sqrt[3]{81} = \sqrt[3]{3^4} = 3^{4/3} = 3 \cdot 3^{1/3} = 3 \cdot \sqrt[3]{3}\)

2. **Calculate \(\sqrt[3]{-192}\)**:
- \(-192 = -1 \cdot 192\)
- Factor \(192\): \(192 = 2^6 \cdot 3\)
- Thus, \(\sqrt[3]{-192} = \sqrt[3]{-1 \cdot 2^6 \cdot 3} = -1 \cdot \sqrt[3]{2^6} \cdot \sqrt[3]{3} = -1 \cdot 2^{6/3} \cdot \sqrt[3]{3} = -2^2 \cdot \sqrt[3]{3} = -4 \cdot \sqrt[3]{3}\)

3. **Calculate \(\sqrt[3]{375}\)**:
- \(375 = 5^3 \cdot 3\)
- Therefore, \(\sqrt[3]{375} = \sqrt[3]{5^3 \cdot 3} = 5 \cdot \sqrt[3]{3}\)

4. **Calculate \(\sqrt[3]{24}\)**:
- \(24 = 2^3 \cdot 3\)
- Thus, \(\sqrt[3]{24} = \sqrt[3]{2^3 \cdot 3} = 2 \cdot \sqrt[3]{3}\)

### Step 2: Substitute back into the expression
Now substituting these values back into the expression:
\[ \frac{(3 \cdot \sqrt[3]{3}) + (-4 \cdot \sqrt[3]{3}) + (5 \cdot \sqrt[3]{3})}{2 \cdot \sqrt[3]{3}} \]

### Step 3: Combine the terms in the numerator
Combine the terms in the numerator:
\[ 3 \cdot \sqrt[3]{3} - 4 \cdot \sqrt[3]{3} + 5 \cdot \sqrt[3]{3} = (3 - 4 + 5) \cdot \sqrt[3]{3} = 4 \cdot \sqrt[3]{3} \]

### Step 4: Simplify the overall expression
Now the expression simplifies to:
\[ \frac{4 \cdot \sqrt[3]{3}}{2 \cdot \sqrt[3]{3}} = \frac{4}{2} = 2 \]

### Final Answer
The value of the expression is \(2\).

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