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### Multiplying Monomials Using the Product Rule:

The **Product Rule** for exponents states:
\[
a^m \cdot a^n = a^{m+n}
\]
This applies when the bases are the same.

#### Numerical Bases:

1. **Problem 17: \( 3^2 \cdot 3^1 \)**
   \[
   = 3^{2+1} = 3^3 = 27
   \]
   
2. **Problem 18: \( (-5)^7 \cdot (-5)^3 \)**
   \[
   = (-5)^{7+3} = (-5)^{10} = 25^{5} = 25^5
   \]

3. **Problem 19: \( 7^2 \cdot 7^5 \)**
   \[
   = 7^{2+5} = 7^7 = 823543
   \]

#### Variable Bases:

4. **Problem 20: \( 12^7 \cdot 12^3 \)**
   \[
   = 12^{7+3} = 12^{10}
   \]
   
5. **Problem 21: \( 5^7 \cdot 5^3 \)**
   \[
   = 5^{7+3} = 5^{10}
   \]

6. **Problem 22: \( (-4)^3 \cdot (-4)^4 \)**
   \[
   = (-4)^{3+4} = (-4)^7
   \]

7. **Problem 23: \( x^{12} \cdot x^3 \)**
   \[
   = x^{12+3} = x^{15}
   \]

8. **Problem 24: \( p^4 \cdot p^7 \)**
   \[
   = p^{4+7} = p^{11}
   \]

9. **Problem 25: \( p^2 \cdot p^6 \)**
   \[
   = p^{2+6} = p^8
   \]

10. **Problem 26: \( x^5 \cdot x^7 \)**
    \[
    = x^{5+7} = x^{12}
    \]

#### Examples with Coefficients:

11. **Problem 27: \( m^2 n^2 \cdot m^3 n^3 \)**
    \[
    = m^{2+3} n^{2+3} = m^5 n^5
    \]

12. **Problem 28: \( x y^2 \cdot x^3 y^2 \)**
    \[
    = x^{1+3} y^{2+2} = x^4 y^4
    \]

Would you like further clarification on any of these steps or additional details? Here are 5 related questions to help you explore more:

1. How does the product rule work when multiplying different bases?
2. Can the product rule be applied to division of exponents? How?
3. What happens when multiplying powers with negative exponents?
4. How does the distributive property differ from the product rule in terms of operations?
5. How do you simplify terms when multiplying multiple variables?

**Tip**: Always double-check that the bases are the same before applying the product rule!

Multiply monomials using the product rule for exponents.

Learn how to multiply monomials using the product rule for exponents. This worksheet provides examples with both numerical and variable bases, including step-by-step solutions for problems like 3^2 * 3^1 and (-5)^7 * (-5)^3.

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