If ` x^(2) +(1)/(x^(2)) = 7. ` find the values of ,
` x - (1)/(x)`

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 7 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x^2 + \frac{1}{x^2} = 7 \]

### Step 2: Relate \( x - \frac{1}{x} \) to \( x^2 + \frac{1}{x^2} \)
We know that:
\[ \left( x - \frac{1}{x} \right)^2 = x^2 - 2 + \frac{1}{x^2} \]
This can be rearranged to:
\[ \left( x - \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} - 2 \]

### Step 3: Substitute the known value
Now, we can substitute \( x^2 + \frac{1}{x^2} \) with 7:
\[ \left( x - \frac{1}{x} \right)^2 = 7 - 2 \]
This simplifies to:
\[ \left( x - \frac{1}{x} \right)^2 = 5 \]

### Step 4: Take the square root
To find \( x - \frac{1}{x} \), we take the square root of both sides:
\[ x - \frac{1}{x} = \pm \sqrt{5} \]

### Final Answer
Thus, the values of \( x - \frac{1}{x} \) are:
\[ x - \frac{1}{x} = \sqrt{5} \quad \text{or} \quad x - \frac{1}{x} = -\sqrt{5} \]

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