If `x + (1)/(x)=4,` then the value of `x ^(4) + (1)/(x ^(4))` is:

Options

A. 124
B. 64
C. 194
D. 81

Answer

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

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + \frac{1}{x} = 4 \]

2. **Square both sides** to find \( x^2 + \frac{1}{x^2} \):
\[ \left( x + \frac{1}{x} \right)^2 = 4^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 16 \]
Therefore, we can rearrange this to find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 16 - 2 = 14 \]

3. **Now square \( x^2 + \frac{1}{x^2} \)** to find \( x^4 + \frac{1}{x^4} \):
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 14^2 \]
This expands to:
\[ x^4 + 2 + \frac{1}{x^4} = 196 \]
Rearranging gives us:
\[ x^4 + \frac{1}{x^4} = 196 - 2 = 194 \]

4. **Final answer**:
\[ x^4 + \frac{1}{x^4} = 194 \]

### Conclusion:
Thus, the value of \( x^4 + \frac{1}{x^4} \) is \( \boxed{194} \).