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Let's work through the expression step by step.

The given expression is:

\[
\frac{x^5(2x)^5}{x^2}
\]

### Step 1: Simplify \( (2x)^5 \)
Use the power rule \((ab)^n = a^n \cdot b^n\):

\[
(2x)^5 = 2^5 \cdot x^5 = 32x^5
\]

### Step 2: Substitute back into the original expression
Now replace \( (2x)^5 \) in the expression:

\[
\frac{x^5(32x^5)}{x^2}
\]

This simplifies to:

\[
\frac{32x^{10}}{x^2}
\]

### Step 3: Simplify the powers of \( x \)
Apply the quotient rule for exponents: \(\frac{x^m}{x^n} = x^{m-n}\):

\[
32x^{10-2} = 32x^8
\]

### Step 4: Compare with \( cx^e \)
We are given that the expression equals \( cx^e \), so by comparison, we have:

- The coefficient \( c \) is \( 32 \),
- The exponent \( e \) is \( 8 \).

### Final Answer:
- The coefficient \( c \) is **32**.
- The exponent \( e \) is **8**.

Would you like more details on any part of this solution? Here are 5 questions that might extend your understanding:

1. How does the power rule \((ab)^n = a^n \cdot b^n\) apply to other expressions?
2. Can we generalize this approach to higher powers and different coefficients?
3. What would happen if the denominator had a more complex power of \( x \)?
4. How does the exponent rule work when dividing terms with variables?
5. How would the solution change if the original expression involved negative exponents?

**Tip:** Always check for simplifications early on to avoid carrying unnecessary complexity throughout the steps!

The expression \( \frac{x^5(2x)^5}{x^2} \) equals \( cx^e \), where the coefficient c is ___, and the exponent e is ___.

Learn how to simplify the expression \( \frac{x^5(2x)^5}{x^2} \) step by step using the power of a product rule and exponent laws. This problem demonstrates how to find the coefficient and exponent in expressions involving powers of variables and constants. Ideal for students in Grades 9-11.

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