`root3(1.728)/root3(13.824)xxroot3(4.096)/root3(216)=`

Options

A. `15/8`
B. `4/15`
C. `2/15`
D. `16/4`

Answer

To solve the expression \( \frac{\sqrt[3]{1.728}}{\sqrt[3]{13.824}} \times \frac{\sqrt[3]{4.096}}{\sqrt[3]{216}} \), we will simplify each cube root step by step.

### Step 1: Simplify Each Cube Root
1. **Calculate \( \sqrt[3]{1.728} \)**:
- We can express 1.728 as \( 1.728 = \frac{1728}{1000} \).
- Since \( 1728 = 12^3 \) and \( 1000 = 10^3 \), we have:
\[ \sqrt[3]{1.728} = \frac{\sqrt[3]{1728}}{\sqrt[3]{1000}} = \frac{12}{10} = 1.2 \]

2. **Calculate \( \sqrt[3]{13.824} \)**:
- We can express 13.824 as \( 13.824 = \frac{13824}{1000} \).
- Since \( 13824 = 24^3 \) and \( 1000 = 10^3 \), we have:
\[ \sqrt[3]{13.824} = \frac{\sqrt[3]{13824}}{\sqrt[3]{1000}} = \frac{24}{10} = 2.4 \]

3. **Calculate \( \sqrt[3]{4.096} \)**:
- We can express 4.096 as \( 4.096 = \frac{4096}{1000} \).
- Since \( 4096 = 16^3 \) and \( 1000 = 10^3 \), we have:
\[ \sqrt[3]{4.096} = \frac{\sqrt[3]{4096}}{\sqrt[3]{1000}} = \frac{16}{10} = 1.6 \]

4. **Calculate \( \sqrt[3]{216} \)**:
- Since \( 216 = 6^3 \), we have:
\[ \sqrt[3]{216} = 6 \]

### Step 2: Substitute Back into the Expression
Now substituting the simplified values back into the expression:
\[ \frac{\sqrt[3]{1.728}}{\sqrt[3]{13.824}} \times \frac{\sqrt[3]{4.096}}{\sqrt[3]{216}} = \frac{1.2}{2.4} \times \frac{1.6}{6} \]

### Step 3: Simplify the Expression
1. **Calculate \( \frac{1.2}{2.4} \)**:
\[ \frac{1.2}{2.4} = \frac{1.2 \div 1.2}{2.4 \div 1.2} = \frac{1}{2} = 0.5 \]

2. **Calculate \( \frac{1.6}{6} \)**:
\[ \frac{1.6}{6} = \frac{1.6 \div 1.6}{6 \div 1.6} = \frac{1}{3.75} \approx 0.267 \]

3. **Combine the Results**:
\[ 0.5 \times 0.267 = 0.1335 \]

### Final Result
Thus, the final result of the expression is approximately \( 0.1335 \).