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We are given the equation:

\[
8^x + 8^x = 1
\]

### Step 1: Simplify the left-hand side
Notice that both terms on the left-hand side are the same, \( 8^x \). So, we can combine them:

\[
8^x + 8^x = 2 \cdot 8^x
\]

The equation becomes:

\[
2 \cdot 8^x = 1
\]

### Step 2: Solve for \( 8^x \)
Now, divide both sides of the equation by 2:

\[
8^x = \frac{1}{2}
\]

### Step 3: Express \( 8 \) as a power of 2
We know that \( 8 = 2^3 \), so we can rewrite the equation as:

\[
(2^3)^x = \frac{1}{2}
\]

Which simplifies to:

\[
2^{3x} = 2^{-1}
\]

### Step 4: Solve for \( x \)
Since the bases are the same (both are powers of 2), we can equate the exponents:

\[
3x = -1
\]

Now, solve for \( x \):

\[
x = -\frac{1}{3}
\]

### Final Answer:
\[
x = -\frac{1}{3}
\]

Would you like more details or have any further questions?

Here are five related questions to expand on this topic:
1. How would the equation change if the base were 9 instead of 8?
2. What happens if the exponent is in a more complex form, like \( 8^{2x} + 8^x = 1 \)?
3. How can you apply logarithms to solve similar equations?
4. What if the equation were \( 8^x + 8^{-x} = 1 \)? How would you approach that?
5. What techniques can be used to solve equations involving different bases (e.g., \( 8^x + 5^x = 1 \))?

**Tip:** When solving exponential equations, converting the bases to the same base (like powers of 2 or 10) can often simplify the process significantly.

Learn how to solve the exponential equation 8^x + 8^x = 1 with a step-by-step solution. This problem focuses on simplifying and solving exponential equations using properties of exponents. Suitable for students in Grades 10-12.

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