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We are tasked with simplifying the expression \(\frac{(15^4 \cdot 7^6)^3}{15^2}\).

Let's break it down step-by-step:

### Step 1: Apply the power rule
The expression inside the parentheses is \((15^4 \cdot 7^6)\), and it is raised to the power of 3. Using the power of a product rule \((a \cdot b)^n = a^n \cdot b^n\), we can distribute the exponent 3 to both bases:

\[
(15^4 \cdot 7^6)^3 = 15^{4 \cdot 3} \cdot 7^{6 \cdot 3} = 15^{12} \cdot 7^{18}
\]

### Step 2: Divide by \(15^2\)
Now, we divide this by \(15^2\):

\[
\frac{15^{12} \cdot 7^{18}}{15^2} = 15^{12-2} \cdot 7^{18} = 15^{10} \cdot 7^{18}
\]

### Final Expression:
\[
15^{10} \cdot 7^{18}
\]

This matches option **d** in the provided answer choices.

### Correct Answer:
The correct expression is **d. \(15^{10} \cdot 7^{18}\)**.

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Would you like more details or have any questions about the steps? 

Here are some related questions to further explore:

1. How do you apply the power of a product rule in different types of problems?
2. What is the general rule for dividing exponents with the same base?
3. How can we simplify an expression where the bases are different but the exponents are the same?
4. What is the importance of knowing exponent rules in algebraic manipulation?
5. Can you derive a similar result with variables instead of numbers?

**Tip:** Always simplify the powers before multiplying or dividing when dealing with expressions involving exponents!

Which expression is equivalent to \(\frac{(15^4 \cdot 7^6)^3}{15^2}\)?

Discover how to simplify the exponential expression \(\frac{(15^4 \cdot 7^6)^3}{15^2}\) using properties of exponents. This algebraic problem is perfect for students in Grades 8-10 and covers key exponent rules like the power of a power and division of exponents.

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