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To solve the equation:

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

we start by expressing \(16\) and \(64\) as powers of \(2\):
- \(16 = 2^4\)
- \(64 = 2^6\)

Rewriting the equation with these bases:
\[
(2^4)^{3x - 3} = (2^6)^{x + 2}
\]

Applying the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[
2^{4(3x - 3)} = 2^{6(x + 2)}
\]

Now, simplify the exponents:
\[
2^{12x - 12} = 2^{6x + 12}
\]

Since the bases are the same, we can set the exponents equal to each other:
\[
12x - 12 = 6x + 12
\]

Now, solve for \(x\):
1. Subtract \(6x\) from both sides:
   \[
   6x - 12 = 12
   \]
2. Add \(12\) to both sides:
   \[
   6x = 24
   \]
3. Divide by \(6\):
   \[
   x = 4
   \]

The solution set is:
\[
\{4\}
\]

Would you like a more detailed breakdown or have any questions?

Here are some related questions you might find interesting:
1. How do you solve exponential equations with different bases?
2. What is the power of a power rule in exponents?
3. How do you convert exponential forms to make the bases the same?
4. Can you solve for \(x\) in exponential equations where the exponents contain multiple variables?
5. What are some other ways to solve exponential equations besides converting to the same base?

**Tip:** When solving exponential equations, converting to a common base can simplify the problem and make it easier to solve.

Solve the equation 16^(3x - 3) = 64^(x + 2)

Learn how to solve the exponential equation 16^(3x - 3) = 64^(x + 2) by expressing both numbers as powers of 2. This detailed breakdown covers exponent rules and solving techniques for high school-level math. Ideal for students in Grades 9-12.

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