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The given equation is:

\[
8 \cdot 0.25^{2x - 3} = 2^{x - 1}
\]

### Step 1: Express \(0.25\) as a power of 2.
Since \(0.25 = \frac{1}{4} = 2^{-2}\), we can rewrite the equation as:

\[
8 \cdot (2^{-2})^{2x - 3} = 2^{x - 1}
\]

### Step 2: Simplify the exponents.
Using the power of a power rule, \((a^m)^n = a^{m \cdot n}\), we simplify:

\[
8 \cdot 2^{-2(2x - 3)} = 2^{x - 1}
\]

This becomes:

\[
8 \cdot 2^{-4x + 6} = 2^{x - 1}
\]

### Step 3: Express \(8\) as a power of 2.
We know that \(8 = 2^3\), so we substitute this:

\[
2^3 \cdot 2^{-4x + 6} = 2^{x - 1}
\]

### Step 4: Combine the powers of 2 on the left-hand side.
Using the rule \(a^m \cdot a^n = a^{m + n}\), we combine the exponents:

\[
2^{3 + (-4x + 6)} = 2^{x - 1}
\]

Simplifying the exponent on the left:

\[
2^{9 - 4x} = 2^{x - 1}
\]

### Step 5: Set the exponents equal to each other.
Since the bases are the same (both are powers of 2), we can equate the exponents:

\[
9 - 4x = x - 1
\]

### Step 6: Solve for \(x\).
Rearrange the equation to solve for \(x\):

\[
9 + 1 = x + 4x
\]
\[
10 = 5x
\]
\[
x = 2
\]

### Final Answer:
The solution is \(x = 2\).

Let me know if you'd like more details or further explanations!

### Related Questions:
1. How do you solve exponential equations when the bases are different?
2. What are the laws of exponents and how are they applied in equations?
3. How can we use logarithms to solve more complex exponential equations?
4. Why is it useful to express numbers like 0.25 as powers of 2 in exponential equations?
5. How can we check the solution of an exponential equation?

### Tip:
Always remember to check your final solution by substituting it back into the original equation to ensure both sides are equal.

Solve the equation 8 * 0.25^(2x - 3) = 2^(x - 1)

Learn how to solve the exponential equation 8 * 0.25^(2x - 3) = 2^(x - 1) with step-by-step instructions. This problem covers key algebraic concepts, including laws of exponents, and is suitable for students in Grades 9-12.

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