`root3 (1- 91/(216))` is equal to-

Options

A. `1/6`
B. `5/6`
C. `1- root3(91)/6`
D. None of these

Answer

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

### Step 1: Simplify the expression inside the cube root
We start with the expression:
\[ 1 - \frac{91}{216} \]
To perform the subtraction, we need to express 1 as a fraction with a denominator of 216:
\[ 1 = \frac{216}{216} \]
Now, we can rewrite the expression:
\[ \frac{216}{216} - \frac{91}{216} = \frac{216 - 91}{216} \]

### Step 2: Calculate the numerator
Now, we calculate \( 216 - 91 \):
\[ 216 - 91 = 125 \]
Thus, we have:
\[ \frac{216 - 91}{216} = \frac{125}{216} \]

### Step 3: Substitute back into the cube root
Now we substitute back into the cube root:
\[ \sqrt[3]{1 - \frac{91}{216}} = \sqrt[3]{\frac{125}{216}} \]

### Step 4: Simplify the cube root
We can separate the cube root of the fraction:
\[ \sqrt[3]{\frac{125}{216}} = \frac{\sqrt[3]{125}}{\sqrt[3]{216}} \]
Now we calculate the cube roots:
- \( \sqrt[3]{125} = 5 \) (since \( 5^3 = 125 \))
- \( \sqrt[3]{216} = 6 \) (since \( 6^3 = 216 \))

Thus, we have:
\[ \frac{\sqrt[3]{125}}{\sqrt[3]{216}} = \frac{5}{6} \]

### Final Answer
Therefore, the value of \( \sqrt[3]{1 - \frac{91}{216}} \) is:
\[ \frac{5}{6} \]

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