`root(3)(1-91/216)` =?

Options

A. (1-5/6)
B. (5/6)
C. `1-sqrt(91)/6`
D. none of these

Answer

To solve the question \( \sqrt[3]{1 - \frac{91}{216}} \), we will follow these steps:

### Step-by-Step Solution:

1. **Write the expression as a single fraction:**
\[ 1 - \frac{91}{216} = \frac{216}{216} - \frac{91}{216} = \frac{216 - 91}{216} \]

2. **Calculate the numerator:**
\[ 216 - 91 = 125 \]
Thus, we have:
\[ 1 - \frac{91}{216} = \frac{125}{216} \]

3. **Substitute back into the cube root:**
\[ \sqrt[3]{1 - \frac{91}{216}} = \sqrt[3]{\frac{125}{216}} \]

4. **Apply the property of cube roots:**
\[ \sqrt[3]{\frac{125}{216}} = \frac{\sqrt[3]{125}}{\sqrt[3]{216}} \]

5. **Calculate the cube roots:**
- The cube root of \( 125 \) is \( 5 \) (since \( 5^3 = 125 \)).
- The cube root of \( 216 \) is \( 6 \) (since \( 6^3 = 216 \)).

Therefore:
\[ \sqrt[3]{\frac{125}{216}} = \frac{5}{6} \]

### Final Answer:
\[ \sqrt[3]{1 - \frac{91}{216}} = \frac{5}{6} \]