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Let's solve the problem step by step.

### Given Expression:
\[ y = \frac{(3^{x+1} - 4) \cdot 3^x}{2 \cdot 3^x} \]

We are asked to find \( 64y \).

### Step 1: Simplify the Expression for \( y \)

The given expression can be simplified as:
\[
y = \frac{3^{x+1} \cdot 3^x - 4 \cdot 3^x}{2 \cdot 3^x}
\]

Since \( 3^{x+1} \cdot 3^x = 3^{x+1+x} = 3^{2x+1} \), the expression for \( y \) becomes:
\[
y = \frac{3^{2x+1} - 4 \cdot 3^x}{2 \cdot 3^x}
\]

We can now factor out \( 3^x \) in the numerator:
\[
y = \frac{3^x(3^{x+1} - 4)}{2 \cdot 3^x}
\]

The \( 3^x \) terms cancel out:
\[
y = \frac{3^{x+1} - 4}{2}
\]

### Step 2: Find \( 64y \)

To find \( 64y \), multiply both sides of the equation by 64:
\[
64y = 64 \cdot \frac{3^{x+1} - 4}{2}
\]

Simplifying:
\[
64y = 32(3^{x+1} - 4)
\]

So, the final expression for \( 64y \) is:
\[
\boxed{64y = 32(3^{x+1} - 4)}
\]

This is the simplified form of the expression.

Would you like more details or have any questions?

Here are some related questions you might explore:
1. How would the expression change if the base of the exponential function were different?
2. What would happen if the constant 4 in the numerator were replaced with another number?
3. How does the expression behave as \( x \) approaches infinity?
4. Can you rewrite the expression for \( y \) using logarithmic functions?
5. What if the entire expression were divided by another function of \( x \)?

**Tip:** When simplifying expressions, always look for common factors that can cancel out across the numerator and denominator to make the expression more manageable.

If y = (3^(x+1) - 4) * 3^x / (2 * 3^x), then find 64y.

Learn how to solve an algebraic problem involving exponential functions and simplification. The problem asks to find the value of 64y given the expression y = (3^(x+1) - 4) * 3^x / (2 * 3^x). Suitable for high school students in Grades 9-12.

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