https://file.aiquickdraw.com/mathbot/file/1727571895426hfmzrx5q.jpg

Let's simplify the given expression:

\[
\frac{3z^4 \cdot 4z^9}{8z^{10}}
\]

### Step-by-step:

1. First, simplify the constants: 
   \[
   \frac{3 \cdot 4}{8} = \frac{12}{8} = \frac{3}{2}
   \]

2. For the powers of \(z\), use the property of exponents: 
   \[
   z^a \cdot z^b = z^{a + b}
   \]
   Applying this:
   \[
   z^4 \cdot z^9 = z^{4 + 9} = z^{13}
   \]

3. Now, the expression is:
   \[
   \frac{3z^{13}}{2z^{10}}
   \]

4. Use the quotient rule for exponents:
   \[
   \frac{z^a}{z^b} = z^{a - b}
   \]
   Applying this:
   \[
   \frac{z^{13}}{z^{10}} = z^{13 - 10} = z^3
   \]

### Final result:
\[
\frac{3}{2} z^3
\]

This is the simplified expression. 

Would you like further details or explanations?

Here are some related questions:
1. How do we apply the quotient rule for exponents?
2. What happens if the exponents are negative?
3. How would the answer change if the exponents were fractional?
4. Can you explain how to handle multiplication and division of terms with different bases?
5. What are some common mistakes made when simplifying expressions with exponents?

**Tip:** Always simplify constants separately from variables to avoid confusion.

Simplify the expression completely: (3z^4 * 4z^9) / 8z^10

Learn how to simplify the algebraic expression (3z^4 * 4z^9) / 8z^10 using exponent rules. Follow a detailed step-by-step solution covering the product of powers, simplification, and quotient of powers. Suitable for Grades 9-12.

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