Which of the following expressions is same as `(1)/((root(3)(2)-1))` ?

Options

A. `root(3)(2)+1`
B. `root(3)(4)+1`
C. `root(3)(4)+root(3)(2)+1`
D. `root(3)(4)+2root(3)(2)+1`

Answer

To solve the expression \( \frac{1}{\sqrt[3]{2} - 1} \), we will rationalize the denominator.

### Step-by-Step Solution:

1. **Identify the expression**:
We start with the expression \( \frac{1}{\sqrt[3]{2} - 1} \).

2. **Multiply by the conjugate**:
To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt[3]{4} + \sqrt[3]{2} + 1 \). This gives us:
\[ \frac{1 \cdot (\sqrt[3]{4} + \sqrt[3]{2} + 1)}{(\sqrt[3]{2} - 1)(\sqrt[3]{4} + \sqrt[3]{2} + 1)} \]

3. **Simplify the denominator**:
The denominator can be simplified using the identity \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, let \( a = \sqrt[3]{2} \) and \( b = 1 \):
\[ (\sqrt[3]{2})^3 - 1^3 = 2 - 1 = 1 \]
Thus, the denominator simplifies to \( 1 \).

4. **Final expression**:
Now, our expression simplifies to:
\[ \sqrt[3]{4} + \sqrt[3]{2} + 1 \]

5. **Conclusion**:
Therefore, the expression \( \frac{1}{\sqrt[3]{2} - 1} \) is equivalent to \( \sqrt[3]{4} + \sqrt[3]{2} + 1 \).