If a `+ (1)/(a) = 2, ` find
` (a^(4) + 1)/(a^(2))`

Answer

To solve the equation \( a + \frac{1}{a} = 2 \) and find \( \frac{a^4 + 1}{a^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ a + \frac{1}{a} = 2 \]
We will square both sides:
\[ \left(a + \frac{1}{a}\right)^2 = 2^2 \]
This simplifies to:
\[ a^2 + 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 4 \]
Which simplifies to:
\[ a^2 + 2 + \frac{1}{a^2} = 4 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} = 4 - 2 \]
This gives us:
\[ a^2 + \frac{1}{a^2} = 2 \]

### Step 3: Find \( a^4 + 1 \)
Next, we need to find \( a^4 + 1 \). We can use the identity:
\[ a^4 + 1 = (a^2)^2 + 1 \]
We can express \( a^4 + 1 \) in terms of \( a^2 + \frac{1}{a^2} \):
\[ a^4 + 1 = a^4 + \frac{1}{a^2} \cdot a^2 = a^4 + a^2 \cdot \frac{1}{a^2} \]
Using the identity \( a^2 + \frac{1}{a^2} = 2 \), we can find \( a^4 + 1 \):
\[ a^4 + 1 = (a^2 + \frac{1}{a^2})^2 - 2 \cdot a^2 \cdot \frac{1}{a^2} \]
Substituting \( a^2 + \frac{1}{a^2} = 2 \):
\[ = 2^2 - 2 = 4 - 2 = 2 \]

### Step 4: Find \( \frac{a^4 + 1}{a^2} \)
Now, we can find \( \frac{a^4 + 1}{a^2} \):
\[ \frac{a^4 + 1}{a^2} = \frac{2}{a^2} \]
Since we know \( a^2 + \frac{1}{a^2} = 2 \), we can find \( a^2 \):
\[ a^2 = 1 \quad \text{(since } a + \frac{1}{a} = 2 \text{ implies } a = 1 \text{)} \]
Thus:
\[ \frac{2}{1} = 2 \]

### Final Answer
Therefore, the value of \( \frac{a^4 + 1}{a^2} \) is:
\[ \boxed{2} \]