Given the equation \( a + \frac{1}{a} = 8 \), we need to find the value of \( a^4 + \frac{1}{a^4} \).

First, we start by squaring both sides of the given equation to find an expression for \( a^2 + \frac{1}{a^2} \).

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

\[
a^2 + 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 64
\]

Since \( a \cdot \frac{1}{a} = 1 \), the equation simplifies to:

\[
a^2 + 2 + \frac{1}{a^2} = 64
\]

Subtracting 2 from both sides, we get:

\[
a^2 + \frac{1}{a^2} = 62
\]

Next, we square \( a^2 + \frac{1}{a^2} \) to find \( a^4 + \frac{1}{a^4} \).

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

\[
a^4 + 2 \cdot a^2 \cdot \frac{1}{a^2} + \frac{1}{a^4} = 3844
\]

Again, since \( a^2 \cdot \frac{1}{a^2} = 1 \), the equation simplifies to:

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

Subtracting 2 from both sides, we get:

\[
a^4 + \frac{1}{a^4} = 3842
\]

Thus, the value of \( a^4 + \frac{1}{a^4} \) is \( \boxed{3842} \).

Would you like any further details or have any questions?

Here are 5 additional questions you might ask:
1. How do you solve quadratic equations?
2. What is the process for finding the derivative of a function?
3. How do you integrate a polynomial function?
4. Can you explain the Pythagorean theorem?
5. What are the properties of logarithms?

**Tip:** When solving algebraic equations, always simplify the equation step by step and check your work to avoid errors.

Discover how to find the value of a^4 + 1/a^4 using the equation a + 1/a = 8. This problem involves algebraic manipulation and understanding of exponents. Suitable for high school students.

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