Let's break down the problem step by step:

We are solving the equation:  
\[
(4^2)^x = 64
\]

### Step 1: Simplify \( (4^2)^x \)
First, look at \( 4^2 \). What does that mean? It's just 4 multiplied by itself:
\[
4^2 = 4 \times 4 = 16
\]

So now, our equation becomes:
\[
16^x = 64
\]

### Step 2: Rewriting 64 as a power of 16
To solve this equation, we need to express both sides with the same base. Let's rewrite 64 in terms of a power of 16. 

Notice that:
\[
16^1 = 16
\]
and
\[
16^2 = 256
\]
That doesn't work. But what if we rewrite 64 using powers of 2?

### Step 3: Rewriting 64 and 16 as powers of 2
- \( 16 = 2^4 \)
- \( 64 = 2^6 \)

Now our equation becomes:
\[
(2^4)^x = 2^6
\]

### Step 4: Applying the power rule
When you have a power raised to another power, you multiply the exponents. So:
\[
(2^4)^x = 2^{4x}
\]

Now, our equation looks like this:
\[
2^{4x} = 2^6
\]

### Step 5: Equating the exponents
Since the bases are the same, we can set the exponents equal to each other:
\[
4x = 6
\]

### Step 6: Solve for \( x \)
Now, divide both sides of the equation by 4 to solve for \( x \):
\[
x = \frac{6}{4} = \frac{3}{2}
\]

So, the solution is:
\[
x = \frac{3}{2}
\]

### Summary:
The solution to \( (4^2)^x = 64 \) is \( x = \frac{3}{2} \).

---

Do you want more details on any step, or have any questions?

Here are 5 related questions you might want to explore:
1. How would you solve \( (3^2)^x = 81 \)?
2. Can you rewrite 128 as a power of 2?
3. How do you simplify powers of powers, like \( (a^b)^c \)?
4. What happens if the base is a fraction?
5. What is a general method to solve exponential equations?

**Tip**: Whenever you're stuck with an exponent equation, try to express both sides using the same base!

Learn how to solve the exponential equation (4^2)^x = 64 with easy-to-follow steps. This problem covers exponents, powers, and the power rule, making it perfect for students in Grades 6-8.

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