We are tasked with solving the equation:

\[
16^{(x+3)} = 64^{(2x-5)}
\]

### Step 1: Express the bases as powers of 2
We can rewrite both 16 and 64 as powers of 2:
- \( 16 = 2^4 \)
- \( 64 = 2^6 \)

Substitute these into the original equation:

\[
(2^4)^{x+3} = (2^6)^{2x-5}
\]

### Step 2: Apply the power of a power rule
Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify both sides:

\[
2^{4(x+3)} = 2^{6(2x-5)}
\]

Now simplify the exponents:

\[
2^{4x + 12} = 2^{12x - 30}
\]

### Step 3: Set the exponents equal
Since the bases are the same, we can equate the exponents:

\[
4x + 12 = 12x - 30
\]

### Step 4: Solve for \(x\)
Now, solve for \(x\):

\[
12 + 30 = 12x - 4x
\]

\[
42 = 8x
\]

\[
x = \frac{42}{8} = 5.25
\]

Thus, the solution is:

\[
x = 5.25
\]

Would you like further clarification or details? 

Here are 5 related questions you can explore:
1. How do you solve exponential equations where the bases are different?
2. What is the general rule for solving \((a^m)^n = a^{m \cdot n}\)?
3. How can logarithms help in solving exponential equations?
4. What are some other methods to simplify powers and exponents?
5. What if the original equation had fractional or negative exponents?

**Tip:** Always check if you can rewrite the bases in exponential equations as powers of the same number. It simplifies the problem significantly!

Explore how to solve the exponential equation 16^(x+3) = 64^(2x-5) using the power of a power rule and equating exponents. This detailed step-by-step solution helps you understand key algebraic concepts and is suitable for students in Grades 9-12.

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