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 values of \( x + \frac{1}{x} \), we can follow these steps:

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

### Step 2: Substitute the given value
We are given that \( x^2 + \frac{1}{x^2} = 7 \). Substituting this into our equation gives:
\[ 7 = \left( x + \frac{1}{x} \right)^2 - 2 \]

### Step 3: Solve for \( \left( x + \frac{1}{x} \right)^2 \)
Adding 2 to both sides, we have:
\[ 7 + 2 = \left( x + \frac{1}{x} \right)^2 \]
\[ 9 = \left( x + \frac{1}{x} \right)^2 \]

### Step 4: Take the square root
Taking the square root of both sides gives:
\[ x + \frac{1}{x} = \pm 3 \]

### Final Answer
Thus, the values of \( x + \frac{1}{x} \) are:
\[ x + \frac{1}{x} = 3 \quad \text{or} \quad x + \frac{1}{x} = -3 \]

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