If `a^(2)+1/(a^(2))=7`, find the value of
`(a+1/(a))`

Answer

To solve the problem, we need to find the value of \( a + \frac{1}{a} \) given that \( a^2 + \frac{1}{a^2} = 7 \).

### Step-by-Step Solution:

1. **Recall the Algebraic Identity**:
We know that:
\[ \left(a + \frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \]
This can be rearranged to express \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} = \left(a + \frac{1}{a}\right)^2 - 2 \]

2. **Substitute the Given Value**:
We are given that:
\[ a^2 + \frac{1}{a^2} = 7 \]
Substitute this into the rearranged identity:
\[ 7 = \left(a + \frac{1}{a}\right)^2 - 2 \]

3. **Solve for \( \left(a + \frac{1}{a}\right)^2 \)**:
Add 2 to both sides:
\[ 7 + 2 = \left(a + \frac{1}{a}\right)^2 \]
\[ 9 = \left(a + \frac{1}{a}\right)^2 \]

4. **Take the Square Root**:
Now, take the square root of both sides:
\[ a + \frac{1}{a} = \sqrt{9} \]
\[ a + \frac{1}{a} = 3 \quad \text{(since we consider only the positive root)} \]

5. **Final Answer**:
Thus, the value of \( a + \frac{1}{a} \) is:
\[ \boxed{3} \]