`root3(4(12)/125)` equals

Options

A. `1(3)/5`
B. `1(2)/5`
C. `7(1)/5`
D. `7(2)/3`

Answer

To solve the problem \( \sqrt[3]{\frac{4 \cdot 12}{125}} \), we can follow these steps:

### Step 1: Simplify the expression inside the cube root
First, we need to calculate the numerator \( 4 \cdot 12 \):
\[ 4 \cdot 12 = 48 \]
So, we rewrite the expression:
\[ \sqrt[3]{\frac{48}{125}} \]

### Step 2: Factor the numerator and the denominator
Next, we can factor both the numerator and the denominator:
- The number \( 48 \) can be factored as \( 16 \cdot 3 \) or \( 4 \cdot 12 \) or \( 2^4 \cdot 3 \).
- The number \( 125 \) can be factored as \( 5^3 \).

Thus, we can express \( 48 \) as:
\[ 48 = 2^4 \cdot 3 \]
So we have:
\[ \sqrt[3]{\frac{2^4 \cdot 3}{5^3}} \]

### Step 3: Rewrite the cube root
We can separate the cube root of the numerator and the denominator:
\[ \sqrt[3]{\frac{2^4 \cdot 3}{5^3}} = \frac{\sqrt[3]{2^4 \cdot 3}}{\sqrt[3]{5^3}} = \frac{\sqrt[3]{2^4} \cdot \sqrt[3]{3}}{5} \]

### Step 4: Simplify \( \sqrt[3]{2^4} \)
We know that \( 2^4 = 2^3 \cdot 2 = 8 \cdot 2 \), thus:
\[ \sqrt[3]{2^4} = \sqrt[3]{8 \cdot 2} = \sqrt[3]{8} \cdot \sqrt[3]{2} = 2 \cdot \sqrt[3]{2} \]

### Step 5: Combine the results
Now, substituting back, we have:
\[ \frac{2 \cdot \sqrt[3]{2} \cdot \sqrt[3]{3}}{5} = \frac{2 \cdot \sqrt[3]{6}}{5} \]

### Step 6: Final expression
Thus, we can express the final result as:
\[ \frac{2 \cdot \sqrt[3]{6}}{5} \]

### Step 7: Convert to mixed number (if needed)
If we need to express this as a mixed number, we can calculate:
\[ \sqrt[3]{6} \approx 1.817 \quad \text{(approximately)} \]
Thus,
\[ \frac{2 \cdot 1.817}{5} \approx \frac{3.634}{5} \approx 0.7268 \]
This can be expressed as \( 1 \frac{3}{5} \) in terms of a mixed number.

### Final Answer
The final answer is:
\[ \sqrt[3]{\frac{48}{125}} = \frac{2 \cdot \sqrt[3]{6}}{5} \approx 1 \frac{3}{5} \]