https://file.aiquickdraw.com/mathbot/file/1727434665555mmg24id0.jpg

The equation provided in the image is:

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

### Steps to solve the equation:

1. **Express 0.25 as a power of 2:**

\[
0.25 = \frac{1}{4} = 2^{-2}
\]

Thus, the equation becomes:

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

2. **Simplify the exponent on the left-hand side:**

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

3. **Express 8 as a power of 2:**

\[
8 = 2^3
\]

Now the equation becomes:

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

4. **Combine the exponents on the left-hand side:**

\[
2^{3 + (-4x + 6)} = 2^{x - 1}
\]
\[
2^{9 - 4x} = 2^{x - 1}
\]

5. **Set the exponents equal to each other (since the bases are the same):**

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

6. **Solve for \(x\):**

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

### Final Answer:
\[
x = 2
\]

Would you like more details on any step or further explanation?

### Related Questions:
1. How do you simplify expressions involving exponents?
2. How do you convert decimals to powers of 2 in an equation?
3. What are the rules for combining like terms with exponents?
4. How does equating exponents work when the bases are the same?
5. What are some methods to solve exponential equations without logarithms?

### Tip:
When dealing with equations involving exponents, try to express all terms with the same base for easier comparison and simplification.

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) step by step. This problem demonstrates key concepts such as expressing decimals as powers of 2 and using exponent laws. Suitable for high school students in Grades 9-12.

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