`root3(4 12/125)`=?

Options

A. `1 2/5`
B. `1 3/5`
C. `1 4/5`
D. `2 2/5`

Answer

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

### Step 1: Convert the mixed number to an improper fraction
First, we need to convert \( 4 \frac{12}{125} \) into an improper fraction.

\[ 4 \frac{12}{125} = \frac{4 \times 125 + 12}{125} = \frac{500 + 12}{125} = \frac{512}{125} \]

### Step 2: Write the cube root expression
Now we can rewrite the original expression using the improper fraction we calculated:

\[ \sqrt[3]{4 \frac{12}{125}} = \sqrt[3]{\frac{512}{125}} \]

### Step 3: Separate the cube root of the numerator and denominator
Using the property of cube roots, we can separate the cube root of the fraction:

\[ \sqrt[3]{\frac{512}{125}} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}} \]

### Step 4: Calculate the cube roots
Now we need to find the cube roots of 512 and 125:

- \( 512 = 8^3 \) so \( \sqrt[3]{512} = 8 \)
- \( 125 = 5^3 \) so \( \sqrt[3]{125} = 5 \)

Thus, we have:

\[ \frac{\sqrt[3]{512}}{\sqrt[3]{125}} = \frac{8}{5} \]

### Step 5: Final answer
Therefore, the final answer is:

\[ \sqrt[3]{4 \frac{12}{125}} = \frac{8}{5} \]