Given that:

$a + \frac{1}{a} = 8$

we are asked to find $a^4 + \frac{1}{a^4}$.

Step-by-Step Solution

1. Square both sides of $a + \frac{1}{a} = 8$:

   $\left(a + \frac{1}{a}\right)^2 = 8^2$

   Expanding the left side:

   $a^2 + 2 + \frac{1}{a^2} = 64$

   Subtracting 2 from both sides:

   $a^2 + \frac{1}{a^2} = 62$

2. Square both sides of $a^2 + \frac{1}{a^2} = 62$ to find $a^4 + \frac{1}{a^4}$:

   $\left(a^2 + \frac{1}{a^2}\right)^2 = 62^2$

   Expanding the left side:

   $a^4 + 2 + \frac{1}{a^4} = 3844$

   Subtracting 2 from both sides:

   $a^4 + \frac{1}{a^4} = 3842$

Final Answer:

$a^4 + \frac{1}{a^4} = 3842$

Would you like further details on any step, or do you have more questions?

---

Here are five related questions for practice:

1. If $a + \frac{1}{a} = 5$, what is $a^2 + \frac{1}{a^2}$?
2. Find $a^3 + \frac{1}{a^3}$ given $a + \frac{1}{a} = 6$.
3. If $a + \frac{1}{a} = 10$, determine $a^4 + \frac{1}{a^4}$.
4. Given $a + \frac{1}{a} = 3$, calculate ( a^2 + \