If x is real and `x^(4)+(1)/(x^(4))=119`, then value of `(x-(1)/(x))` is

Options

A. `pm 4`
B. `pm 9`
C. `pm 3`
D. `pm 2`

Answer

To solve the equation \( x^4 + \frac{1}{x^4} = 119 \) 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:
\[ x^4 + \frac{1}{x^4} = 119 \]

### Step 2: Add 2 to both sides
We can add 2 to both sides:
\[ x^4 + \frac{1}{x^4} + 2 = 121 \]

### Step 3: Recognize the identity
Notice that:
\[ x^4 + \frac{1}{x^4} + 2 = \left(x^2 + \frac{1}{x^2}\right)^2 \]
Thus, we can rewrite the left side:
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 121 \]

### Step 4: Take the square root
Taking the square root of both sides gives:
\[ x^2 + \frac{1}{x^2} = \pm 11 \]

### Step 5: Solve for \( x^2 + \frac{1}{x^2} \)
Since \( x \) is real, we consider the positive case:
\[ x^2 + \frac{1}{x^2} = 11 \]

### Step 6: Use the identity again
Now, we can use the identity:
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \]
So we set:
\[ \left(x - \frac{1}{x}\right)^2 + 2 = 11 \]

### Step 7: Isolate the square
Subtract 2 from both sides:
\[ \left(x - \frac{1}{x}\right)^2 = 9 \]

### Step 8: Take the square root
Taking the square root of both sides gives:
\[ x - \frac{1}{x} = \pm 3 \]

### Final Result
Thus, the value of \( x - \frac{1}{x} \) is:
\[ \boxed{3} \text{ or } \boxed{-3} \]