If `x = 3 + sqrt8`, find the value of `x^(3) + (1)/(x^(3))`.

Options

A. 16
B. 198
C. 200
D. 196

Answer

To find the value of \( x^3 + \frac{1}{x^3} \) given \( x = 3 + \sqrt{8} \), we can follow these steps:

### Step 1: Calculate \( \frac{1}{x} \)
We start by finding \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \cdot \frac{3 - \sqrt{8}}{3 - \sqrt{8}} = \frac{3 - \sqrt{8}}{(3 + \sqrt{8})(3 - \sqrt{8})} \]
Calculating the denominator:
\[ (3 + \sqrt{8})(3 - \sqrt{8}) = 3^2 - (\sqrt{8})^2 = 9 - 8 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 3 - \sqrt{8} \]

### Step 2: Calculate \( x + \frac{1}{x} \)
Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (3 + \sqrt{8}) + (3 - \sqrt{8}) = 3 + \sqrt{8} + 3 - \sqrt{8} = 6 \]

### Step 3: Use the formula for \( x^3 + \frac{1}{x^3} \)
We can use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) \]
Substituting \( x + \frac{1}{x} = 6 \):
\[ x^3 + \frac{1}{x^3} = 6^3 - 3 \times 6 \]
Calculating \( 6^3 \):
\[ 6^3 = 216 \]
Calculating \( 3 \times 6 \):
\[ 3 \times 6 = 18 \]
Now substituting back:
\[ x^3 + \frac{1}{x^3} = 216 - 18 = 198 \]

### Final Answer
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{198} \]