To solve the equation \( 64^{x-1} = 8^3 \), follow these steps:

### Step 1: Express both sides with the same base
Notice that both 64 and 8 can be expressed as powers of 2:

\[
64 = 2^6 \quad \text{and} \quad 8 = 2^3
\]

So, rewrite the equation using base 2:

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

### Step 2: Simplify the equation using the power of a power rule
Apply the power of a power rule, \( (a^m)^n = a^{m \cdot n} \), to both sides:

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

### Step 3: Set the exponents equal
Since the bases are the same, set the exponents equal to each other:

\[
6(x - 1) = 9
\]

### Step 4: Solve for \( x \)
Expand and solve for \( x \):

\[
6x - 6 = 9
\]

Add 6 to both sides:

\[
6x = 15
\]

Now, divide by 6:

\[
x = \frac{15}{6} = \frac{5}{2}
\]

So, the solution is:

\[
\boxed{x = \frac{5}{2}}
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you solve exponential equations with different bases?
2. What is the significance of the power of a power rule in solving exponential equations?
3. How can logarithms be used to solve exponential equations?
4. What are the steps for solving equations involving multiple exponents?
5. How can you verify the solution to an exponential equation?
6. What are the key properties of exponential functions that aid in solving equations?
7. How do exponential equations relate to logarithmic equations?
8. What methods can be used to solve exponential equations graphically?

**Tip:** Always try to express both sides of the equation with the same base when dealing with exponential equations, as it simplifies the process of solving them.

Learn how to solve the exponential equation 64^(x-1) = 8^3 with a detailed step-by-step solution. This problem covers exponential equations, power rules, and logarithms. Suitable for high school students in Grades 10-12.

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