If `x^(4) +(1)/(x^(4)) =2`, and x isa positive quantity, then x is equal to:

Options

A. a) 1
B. b) 2
C. c) `(1)/(2)`
D. d) `(1)/(4)`

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 2 \), we will follow these steps:

### Step 1: Understand the equation
We are given the equation \( x^4 + \frac{1}{x^4} = 2 \). Our goal is to find the value of \( x \).

### Step 2: Analyze the expression
Notice that \( x^4 + \frac{1}{x^4} \) can be rewritten using the identity for squares. We know that for any positive number \( a \):
\[ a + \frac{1}{a} \geq 2 \]
This equality holds when \( a = 1 \).

### Step 3: Set \( x^4 \) to a variable
Let \( y = x^4 \). Then the equation becomes:
\[ y + \frac{1}{y} = 2 \]

### Step 4: Multiply through by \( y \)
To eliminate the fraction, multiply both sides by \( y \):
\[ y^2 + 1 = 2y \]

### Step 5: Rearrange the equation
Rearranging gives us a quadratic equation:
\[ y^2 - 2y + 1 = 0 \]

### Step 6: Factor the quadratic
This can be factored as:
\[ (y - 1)^2 = 0 \]

### Step 7: Solve for \( y \)
Setting the factor equal to zero gives:
\[ y - 1 = 0 \implies y = 1 \]

### Step 8: Substitute back for \( x \)
Recall that \( y = x^4 \). Therefore:
\[ x^4 = 1 \]

### Step 9: Solve for \( x \)
Taking the fourth root of both sides, we find:
\[ x = 1 \]
Since \( x \) is specified to be a positive quantity, we take the positive root.

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{1} \]