If `(x^(4)+(1)/(x^(4)))=34`, then the value of `(x-(1)/(x))` is :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 34 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^4 + \frac{1}{x^4} = 34 \]

### Step 2: Add 2 to both sides
We add 2 to both sides of the equation:
\[ x^4 + \frac{1}{x^4} + 2 = 34 + 2 \]
This simplifies to:
\[ x^4 + \frac{1}{x^4} + 2 = 36 \]

### Step 3: Recognize the identity
We can use the identity:
\[ a^2 + b^2 + 2ab = (a + b)^2 \]
In our case, let \( a = x^2 \) and \( b = \frac{1}{x^2} \). Thus, we can rewrite the left side:
\[ (x^2 + \frac{1}{x^2})^2 = 36 \]

### Step 4: Take the square root
Taking the square root of both sides gives us:
\[ x^2 + \frac{1}{x^2} = 6 \quad \text{(since both sides are positive)} \]

### Step 5: Subtract 2 from both sides
Now, we subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} - 2 = 6 - 2 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} - 2 = 4 \]

### Step 6: Recognize another identity
We can again use the identity:
\[ a^2 + b^2 - 2ab = (a - b)^2 \]
Let \( a = x \) and \( b = \frac{1}{x} \). Thus, we can rewrite the left side:
\[ (x - \frac{1}{x})^2 = 4 \]

### Step 7: Take the square root again
Taking the square root of both sides gives us:
\[ x - \frac{1}{x} = 2 \quad \text{or} \quad x - \frac{1}{x} = -2 \]

### Conclusion
The value of \( x - \frac{1}{x} \) can be either 2 or -2. However, since we are looking for a positive value, we take:
\[ \boxed{2} \]