`(3root3(13824))/(2root3(-15625))+(2root3(-13824))/root3(5832)=`

Options

A. `50/27`
B. `-50/27`
C. `27/50`
D. `-27/50`

Answer

To solve the expression \((3\sqrt[3]{13824})/(2\sqrt[3]{-15625}) + (2\sqrt[3]{-13824})/\sqrt[3]{5832}\), we will break it down step by step.

### Step 1: Calculate the cube roots
First, we need to calculate the cube roots of the numbers involved.

1. **Calculate \(\sqrt[3]{13824}\)**:
- \(13824 = 24^3\)
- Therefore, \(\sqrt[3]{13824} = 24\).

2. **Calculate \(\sqrt[3]{-15625}\)**:
- \(-15625 = -25^3\)
- Therefore, \(\sqrt[3]{-15625} = -25\).

3. **Calculate \(\sqrt[3]{-13824}\)**:
- \(-13824 = -24^3\)
- Therefore, \(\sqrt[3]{-13824} = -24\).

4. **Calculate \(\sqrt[3]{5832}\)**:
- \(5832 = 18^3\)
- Therefore, \(\sqrt[3]{5832} = 18\).

### Step 2: Substitute the cube roots back into the expression
Now we substitute these values back into the expression:

\[ \frac{3 \cdot 24}{2 \cdot (-25)} + \frac{2 \cdot (-24)}{18} \]

### Step 3: Simplify each term
1. **Simplify the first term**:
\[ \frac{3 \cdot 24}{2 \cdot (-25)} = \frac{72}{-50} = -\frac{72}{50} = -\frac{36}{25} \]

2. **Simplify the second term**:
\[ \frac{2 \cdot (-24)}{18} = \frac{-48}{18} = -\frac{8}{3} \]

### Step 4: Combine the two terms
Now we need to combine \(-\frac{36}{25}\) and \(-\frac{8}{3}\). To do this, we need a common denominator.

- The least common multiple of 25 and 3 is 75.

Convert both fractions:
1. **Convert \(-\frac{36}{25}\)**:
\[ -\frac{36}{25} = -\frac{36 \cdot 3}{25 \cdot 3} = -\frac{108}{75} \]

2. **Convert \(-\frac{8}{3}\)**:
\[ -\frac{8}{3} = -\frac{8 \cdot 25}{3 \cdot 25} = -\frac{200}{75} \]

### Step 5: Add the two fractions
Now we can add the two fractions:
\[ -\frac{108}{75} - \frac{200}{75} = -\frac{108 + 200}{75} = -\frac{308}{75} \]

### Final Answer
Thus, the final answer is:
\[ -\frac{308}{75} \]