If `x ^(4) + (1)/(x ^(4))= 98 and x gt 1` then what is the value of `x + (1)/(x)` ?

Options

A. 2
B. `2 sqrt3`
C. `sqrt5`
D. `sqrt3`

Answer

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ x^4 + \frac{1}{x^4} = 98 \]

2. **Introduce a New Variable**:
Let \( y = x^2 + \frac{1}{x^2} \). We know that:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Therefore, we can rewrite the equation as:
\[ y^2 - 2 = 98 \]

3. **Solve for \( y^2 \)**:
Rearranging gives:
\[ y^2 = 98 + 2 = 100 \]

4. **Take the Square Root**:
Taking the square root of both sides, we find:
\[ y = \sqrt{100} = 10 \]
Since \( x > 1 \), \( y \) must be positive.

5. **Relate \( y \) Back to \( x \)**:
Now, we need to find \( x + \frac{1}{x} \). We know:
\[ y = x^2 + \frac{1}{x^2} \]
We can express \( x + \frac{1}{x} \) in terms of \( y \):
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Let \( z = x + \frac{1}{x} \). Then:
\[ y = z^2 - 2 \]

6. **Set Up the Equation**:
Substituting \( y = 10 \):
\[ 10 = z^2 - 2 \]
Rearranging gives:
\[ z^2 = 10 + 2 = 12 \]

7. **Take the Square Root**:
Taking the square root of both sides, we find:
\[ z = \sqrt{12} = 2\sqrt{3} \]
Since \( x > 1 \), we take the positive root.

8. **Final Answer**:
Therefore, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{2\sqrt{3}} \]