If `x^((1)/(4)) + x^((-1)/(4))=2`, then what is the value of `x^(81) + ((1)/(x^(81)))`?

Options

A. `-2`
B. 0
C. 1
D. 2

Answer

To solve the equation \( x^{\frac{1}{4}} + x^{-\frac{1}{4}} = 2 \) and find the value of \( x^{81} + \frac{1}{x^{81}} \), we can follow these steps:

### Step 1: Simplify the Given Equation
Given:
\[ x^{\frac{1}{4}} + x^{-\frac{1}{4}} = 2 \]
Let \( y = x^{\frac{1}{4}} \). Then the equation becomes:
\[ y + \frac{1}{y} = 2 \]

### Step 2: Multiply by \( y \)
Multiply both sides by \( y \):
\[ y^2 + 1 = 2y \]
Rearranging gives:
\[ y^2 - 2y + 1 = 0 \]

### Step 3: Factor the Quadratic Equation
The equation can be factored as:
\[ (y - 1)^2 = 0 \]
Thus, we find:
\[ y - 1 = 0 \implies y = 1 \]

### Step 4: Substitute Back for \( x \)
Since \( y = x^{\frac{1}{4}} \), we have:
\[ x^{\frac{1}{4}} = 1 \]
Raising both sides to the power of 4 gives:
\[ x = 1 \]

### Step 5: Find \( x^{81} + \frac{1}{x^{81}} \)
Now we need to calculate:
\[ x^{81} + \frac{1}{x^{81}} \]
Substituting \( x = 1 \):
\[ 1^{81} + \frac{1}{1^{81}} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( x^{81} + \frac{1}{x^{81}} \) is:
\[ \boxed{2} \]