Find the value of `root3((4913)/(2197))`.

Options

A. `(17)/(13)`
B. `(13)/(17)`
C. `(13)/(7)`
D. `(17)/(7)`

Answer

To find the value of \( \sqrt[3]{\frac{4913}{2197}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt[3]{\frac{4913}{2197}} = \frac{\sqrt[3]{4913}}{\sqrt[3]{2197}} \]

### Step 2: Factor the numbers
Next, we need to factor \( 4913 \) and \( 2197 \) to see if they are perfect cubes.

- **Finding \( 4913 \)**:
- We can try to find the cube root of \( 4913 \).
- \( 17^3 = 17 \times 17 \times 17 = 4913 \).

- **Finding \( 2197 \)**:
- Similarly, we can find the cube root of \( 2197 \).
- \( 13^3 = 13 \times 13 \times 13 = 2197 \).

### Step 3: Substitute the cube roots
Now we can substitute these values back into our expression:
\[ \frac{\sqrt[3]{4913}}{\sqrt[3]{2197}} = \frac{17}{13} \]

### Step 4: Final result
Thus, the value of \( \sqrt[3]{\frac{4913}{2197}} \) is:
\[ \frac{17}{13} \]