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.