If `a^2 + (1)/( a^2 ) = 2`, find : (i) `a+ (1)/( a)`

Answer

To solve the problem, we start with the given equation:

**Given:**
\[ a^2 + \frac{1}{a^2} = 2 \]

We need to find the value of \( a + \frac{1}{a} \).

### Step 1: Use the identity
We know the identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]

In our case, let:
- \( a = a \)
- \( b = \frac{1}{a} \)

Then:
\[ (a + \frac{1}{a})^2 = a^2 + \frac{1}{a^2} + 2 \cdot a \cdot \frac{1}{a} \]

### Step 2: Simplify the expression
The expression simplifies to:
\[ (a + \frac{1}{a})^2 = a^2 + \frac{1}{a^2} + 2 \]

### Step 3: Substitute the known value
We know from the problem statement that:
\[ a^2 + \frac{1}{a^2} = 2 \]

Substituting this into our equation gives:
\[ (a + \frac{1}{a})^2 = 2 + 2 \]

### Step 4: Calculate the right side
Now, calculate the right side:
\[ (a + \frac{1}{a})^2 = 4 \]

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

### Step 6: Final answer
Thus:
\[ a + \frac{1}{a} = 2 \]

### Summary of the solution:
The value of \( a + \frac{1}{a} \) is 2.

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