Find the value of `root(3)(4913/2197)`
(a)17/13
(b)13/17
(c)13/7
(d)17/7

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: Identify the cube roots of the numerator and denominator
We need to find the cube roots of both 4913 and 2197.

- **Numerator**: \( 4913 \)
- **Denominator**: \( 2197 \)

### Step 2: Find the cube root of 4913
Let's check if 4913 is a perfect cube.

Calculating \( 17^3 \):
\[ 17 \times 17 = 289 \]
\[ 289 \times 17 = 4913 \]
Thus, \( 4913 = 17^3 \).

### Step 3: Find the cube root of 2197
Now, let's check if 2197 is a perfect cube.

Calculating \( 13^3 \):
\[ 13 \times 13 = 169 \]
\[ 169 \times 13 = 2197 \]
Thus, \( 2197 = 13^3 \).

### Step 4: Rewrite the expression
Now we can rewrite the original expression:
\[ \sqrt[3]{\frac{4913}{2197}} = \sqrt[3]{\frac{17^3}{13^3}} \]

### Step 5: Simplify the expression
Using the property of cube roots, we can simplify:
\[ \sqrt[3]{\frac{17^3}{13^3}} = \frac{\sqrt[3]{17^3}}{\sqrt[3]{13^3}} = \frac{17}{13} \]

### Step 6: Final answer
Thus, the value of \( \sqrt[3]{\frac{4913}{2197}} \) is:
\[ \frac{17}{13} \]

### Conclusion
The correct answer is option (a) \( \frac{17}{13} \).

---