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The problem given is:

\[
\text{Diketahui } 2^x + 2^{-x} = 3 \text{ dan } 2^x \cdot 2^{-x} = 2.
\]
\[
\text{Nilai dari } 8^x + 8^{-x} \text{ adalah} \ldots
\]

### Step 1: Analyzing the Given Equations

1. **Equation 1**: \( 2^x + 2^{-x} = 3 \)
2. **Equation 2**: \( 2^x \cdot 2^{-x} = 2 \)

### Step 2: Simplifying the Equations
- **Equation 2** simplifies to \( 2^x \cdot 2^{-x} = 1 \) (since \( 2^x \cdot 2^{-x} = 2^{x-x} = 2^0 = 1 \)).
- The equation \( 2 \) given appears to be incorrectly simplified. We should consider the original question which likely has \( 2^x \cdot 2^{-x} = 1 \), as this is the fundamental identity for exponentials.

### Step 3: Working on the Expression \( 8^x + 8^{-x} \)
The expression \( 8^x + 8^{-x} \) can be rewritten using the identity:

\[
8^x = (2^3)^x = (2^x)^3
\]
\[
8^{-x} = (2^{-x})^3 = (2^{-x})^3
\]

Thus, the expression becomes:

\[
8^x + 8^{-x} = (2^x)^3 + (2^{-x})^3
\]

### Step 4: Applying the Identity for Cubes
Using the identity for the sum of cubes:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

Where \( a = 2^x \) and \( b = 2^{-x} \), we already know:

\[
2^x + 2^{-x} = 3
\]

So:

\[
(2^x)^3 + (2^{-x})^3 = (3)((2^x)^2 - 2^x \cdot 2^{-x} + (2^{-x})^2)
\]

### Step 5: Simplifying the Expression
Simplify \( (2^x)^2 + (2^{-x})^2 \):

\[
(2^x)^2 = 2^{2x}, \quad (2^{-x})^2 = 2^{-2x}
\]

So,

\[
8^x + 8^{-x} = 3 \times \left(2^{2x} + 2^{-2x} + 1\right)
\]

Given that \( 2^x + 2^{-x} = 3 \), we can infer \( 2^{2x} + 2^{-2x} + 1 = 7 \).

Thus, we compute:

\[
8^x + 8^{-x} = 3 \times 7 = 21
\]

However, there seems to be a mismatch with the options provided. The process might involve additional simplifications or reconsideration based on the exact identities given.

Let me know if you'd like further details or another explanation. Here are some related questions:

1. How can you express \( 2^{3x} \) in terms of \( 2^x \)?
2. What are the identities involved in expanding \( a^3 + b^3 \)?
3. How does \( 2^x \cdot 2^{-x} \) simplify to 1?
4. What other methods could be used to simplify \( 8^x + 8^{-x} \)?
5. Can you solve \( 2^x + 2^{-x} = 3 \) for \( x \) explicitly?

**Tip:** When dealing with exponential expressions, remember to explore identities involving squares and cubes to simplify complex expressions.

Explore the solution to the problem involving exponential equations: 8^x + 8^-x. This problem delves into advanced concepts of exponential functions and cubic identities, providing step-by-step explanations and formulas used. Suitable for advanced high school students and those preparing for competitive exams.

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