If `x^(4)+(1)/(x^(4))=23` then what is the value of `(x-(1)/(x))^(2)`?

Options

A. `-3`
B. 3
C. 7
D. `-7`

Answer

To solve the problem, we need to find the value of \((x - \frac{1}{x})^2\) given that \(x^4 + \frac{1}{x^4} = 23\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x^4 + \frac{1}{x^4} = 23 \]

2. **Add 2 to both sides:**
\[ x^4 + \frac{1}{x^4} + 2 = 25 \]

3. **Recognize the identity:**
The left side can be rewritten using the identity:
\[ (x^2 + \frac{1}{x^2})^2 = x^4 + 2 + \frac{1}{x^4} \]
Thus, we have:
\[ (x^2 + \frac{1}{x^2})^2 = 25 \]

4. **Take the square root of both sides:**
\[ x^2 + \frac{1}{x^2} = 5 \quad \text{(since we take the positive root)} \]

5. **Now, we need to find \((x - \frac{1}{x})^2\):**
We use the identity:
\[ (x - \frac{1}{x})^2 = (x^2 + \frac{1}{x^2}) - 2 \]

6. **Substituting the value we found:**
\[ (x - \frac{1}{x})^2 = 5 - 2 = 3 \]

### Final Answer:
\[ (x - \frac{1}{x})^2 = 3 \]