Solve for x in the exponential equation: $2^{2x} - 9 \cdot 2^x + 8 = 0$

Step 1: Rewrite the equation

We can rewrite the term $2^{2x}$ as $(2^x)^2$ using the exponent rule $(a^m)^n = a^{mn}$. This helps us see a pattern in the equation.

Step 2: Substitute to form a quadratic equation

To simplify the equation, we can substitute $y$ for $2^x$. This transforms the exponential equation into a standard quadratic equation in terms of $y$.

Step 3: Factor the quadratic equation

We need to find two numbers that multiply to 8 and add up to -9. These numbers are -1 and -8. So, we can factor the quadratic equation as $(y - 1)(y - 8) = 0$.

Step 4: Solve for y

For the product of two factors to be zero, at least one of the factors must be zero. This gives us two possible values for $y$: $y=1$ or $y=8$.

Step 5: Substitute back and solve for x

Now we substitute back $2^x$ for $y$. For $2^x = 1$, we know that any non-zero number raised to the power of 0 is 1, so $x=0$. For $2^x = 8$, since $8 = 2^3$, we have $x=3$.

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