If x = 3 + `sqrt(8) ` find the value of `x^(4) + (1)/(x^(4))`

Options

A. 1254
B. 1064
C. 1154
D. 1206

Answer

To solve the problem, we need to find the value of \( x^4 + \frac{1}{x^4} \) given that \( x = 3 + \sqrt{8} \).

### Step-by-step Solution:

1. **Find \( \frac{1}{x} \)**:
\[ x = 3 + \sqrt{8} \]
To find \( \frac{1}{x} \), we rationalize the denominator:
\[ \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \cdot \frac{3 - \sqrt{8}}{3 - \sqrt{8}} = \frac{3 - \sqrt{8}}{(3 + \sqrt{8})(3 - \sqrt{8})} \]
The denominator simplifies as follows:
\[ (3 + \sqrt{8})(3 - \sqrt{8}) = 3^2 - (\sqrt{8})^2 = 9 - 8 = 1 \]
Therefore,
\[ \frac{1}{x} = 3 - \sqrt{8} \]

2. **Calculate \( x + \frac{1}{x} \)**:
\[ x + \frac{1}{x} = (3 + \sqrt{8}) + (3 - \sqrt{8}) = 6 \]

3. **Square \( x + \frac{1}{x} \)**:
\[ \left( x + \frac{1}{x} \right)^2 = 6^2 = 36 \]
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ x^2 + 2 + \frac{1}{x^2} = 36 \]
Therefore,
\[ x^2 + \frac{1}{x^2} = 36 - 2 = 34 \]

4. **Square \( x^2 + \frac{1}{x^2} \)** to find \( x^4 + \frac{1}{x^4} \)**:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 34^2 = 1156 \]
Again using the identity:
\[ x^4 + 2 + \frac{1}{x^4} = 1156 \]
Thus,
\[ x^4 + \frac{1}{x^4} = 1156 - 2 = 1154 \]

### Final Answer:
\[ x^4 + \frac{1}{x^4} = 1154 \]