If `x^(4)+(1)/(x^(4))=14` then find the value of `x-(1)/(x)`.

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 14 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Relate \( x^4 + \frac{1}{x^4} \) to \( x^2 + \frac{1}{x^2} \)

We know that:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
This means we can express \( x^4 + \frac{1}{x^4} \) in terms of \( x^2 + \frac{1}{x^2} \).

### Step 2: Substitute the given value

Given that:
\[ x^4 + \frac{1}{x^4} = 14 \]
we can substitute this into our equation:
\[ 14 = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]

### Step 3: Solve for \( x^2 + \frac{1}{x^2} \)

Rearranging the equation gives:
\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 14 + 2 = 16 \]
Taking the square root of both sides:
\[ x^2 + \frac{1}{x^2} = \sqrt{16} = 4 \]

### Step 4: Relate \( x^2 + \frac{1}{x^2} \) to \( x - \frac{1}{x} \)

We can use the identity:
\[ x^2 + \frac{1}{x^2} = \left( x - \frac{1}{x} \right)^2 + 2 \]
Substituting the value we found:
\[ 4 = \left( x - \frac{1}{x} \right)^2 + 2 \]

### Step 5: Solve for \( x - \frac{1}{x} \)

Rearranging gives:
\[ \left( x - \frac{1}{x} \right)^2 = 4 - 2 = 2 \]
Taking the square root of both sides:
\[ x - \frac{1}{x} = \sqrt{2} \quad \text{or} \quad x - \frac{1}{x} = -\sqrt{2} \]

### Final Answer

Thus, the value of \( x - \frac{1}{x} \) is:
\[ \sqrt{2} \quad \text{or} \quad -\sqrt{2} \]