`root(3)(1 - 127/343)` is equal to

Options

A. `5/9`
B. `1 -1/7`
C. `4/7`
D. `1 - 2/7`

Answer

To solve the expression \( \sqrt[3]{1 - \frac{127}{343}} \), we can follow these steps:

### Step 1: Find a common denominator
We need to combine the terms inside the cube root. The denominator of the fraction is 343. We can rewrite 1 as \( \frac{343}{343} \) to have a common denominator.

\[ 1 - \frac{127}{343} = \frac{343}{343} - \frac{127}{343} \]

### Step 2: Subtract the fractions
Now that we have a common denominator, we can subtract the fractions:

\[ \frac{343 - 127}{343} = \frac{216}{343} \]

### Step 3: Substitute back into the cube root
Now we can substitute this result back into the cube root:

\[ \sqrt[3]{1 - \frac{127}{343}} = \sqrt[3]{\frac{216}{343}} \]

### Step 4: Simplify the cube root
We can separate the cube root of the numerator and the denominator:

\[ \sqrt[3]{\frac{216}{343}} = \frac{\sqrt[3]{216}}{\sqrt[3]{343}} \]

### Step 5: Calculate the cube roots
Next, we need to calculate the cube roots. We know that:

- \( 216 = 6^3 \), so \( \sqrt[3]{216} = 6 \)
- \( 343 = 7^3 \), so \( \sqrt[3]{343} = 7 \)

Thus, we have:

\[ \frac{\sqrt[3]{216}}{\sqrt[3]{343}} = \frac{6}{7} \]

### Step 6: Final result
Therefore, the final result is:

\[ \sqrt[3]{1 - \frac{127}{343}} = \frac{6}{7} \]

### Summary
The expression \( \sqrt[3]{1 - \frac{127}{343}} \) simplifies to \( \frac{6}{7} \).

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