We are given the equation:

\[
64^{2 - x} = 2^{4x + 3}
\]

### Step 1: Express 64 as a power of 2
We know that \(64 = 2^6\), so we can rewrite the equation as:

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

### Step 2: Apply exponent rules
Using the rule \((a^m)^n = a^{m \cdot n}\), we get:

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

Simplify the exponent on the left-hand side:

\[
2^{12 - 6x} = 2^{4x + 3}
\]

### Step 3: Equate the exponents
Since the bases are the same (both are powers of 2), we can equate the exponents:

\[
12 - 6x = 4x + 3
\]

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

\[
12 - 3 = 4x + 6x
\]
\[
9 = 10x
\]
\[
x = \frac{9}{10}
\]

Thus, the solution is:

\[
x = \frac{9}{10}
\]

Would you like any additional details or explanations?

Here are 5 related questions to consider:
1. How can you simplify equations with exponents further?
2. How do you solve equations involving logarithms?
3. How does the base of an exponent affect the equation-solving process?
4. Can exponential equations have more than one solution?
5. What are the properties of exponents and how are they used in solving equations?

**Tip:** Always simplify the base of an exponent if possible; it often makes solving the equation easier!

Learn how to solve the exponential equation 64^(2 - x) = 2^(4x + 3) by expressing powers of 64 in terms of base 2 and using exponent rules. This step-by-step solution covers key algebraic and exponential concepts, suitable for students in Grades 9-12.

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