If a `+ (1)/(a) = 2, ` find
` (a^(8) +1)/(a^(4))`

Answer

To solve the equation \( a + \frac{1}{a} = 2 \) and find the value of \( \frac{a^8 + 1}{a^4} \), we will follow these steps:

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

### Step 2: Rearrange to find \( a^2 + \frac{1}{a^2} \)
From the equation above, we can rearrange it:
\[ a^2 + \frac{1}{a^2} = 4 - 2 = 2 \]

### Step 3: Square again to find \( a^4 + \frac{1}{a^4} \)
Now we will square \( a^2 + \frac{1}{a^2} \):
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 2^2 \]
This expands to:
\[ a^4 + 2 + \frac{1}{a^4} = 4 \]
Rearranging gives:
\[ a^4 + \frac{1}{a^4} = 4 - 2 = 2 \]

### Step 4: Find \( a^8 + \frac{1}{a^8} \)
Now we square \( a^4 + \frac{1}{a^4} \):
\[ \left(a^4 + \frac{1}{a^4}\right)^2 = 2^2 \]
This expands to:
\[ a^8 + 2 + \frac{1}{a^8} = 4 \]
Rearranging gives:
\[ a^8 + \frac{1}{a^8} = 4 - 2 = 2 \]

### Step 5: Calculate \( \frac{a^8 + 1}{a^4} \)
Now we need to find \( \frac{a^8 + 1}{a^4} \):
\[ \frac{a^8 + 1}{a^4} = \frac{a^8}{a^4} + \frac{1}{a^4} = a^4 + \frac{1}{a^4} \]
From Step 3, we found that:
\[ a^4 + \frac{1}{a^4} = 2 \]
Thus:
\[ \frac{a^8 + 1}{a^4} = 2 \]

### Final Answer
The value of \( \frac{a^8 + 1}{a^4} \) is \( \boxed{2} \).