Exponents Worksheets with Answer Key - Free Printable

Problem Overview:

The task involves using the Product Rule of Exponents to simplify expressions. The Product Rule states:

\[
a^m \times a^n = a^{m+n}
\]

This means when multiplying two powers with the same base, you add their exponents.

Solution:

We will solve each expression step by step using the Product Rule.

---

#### Expression 1: \( 12^{12} \times 12^{15} \)

- Base: \( 12 \)
- Exponents: \( 12 \) and \( 15 \)

Using the Product Rule:
\[
12^{12} \times 12^{15} = 12^{12 + 15} = 12^{27}
\]

Answer: \( 12^{27} \)

---

#### Expression 2: \( 14^{17} \times 14^{20} \)

- Base: \( 14 \)
- Exponents: \( 17 \) and \( 20 \)

Using the Product Rule:
\[
14^{17} \times 14^{20} = 14^{17 + 20} = 14^{37}
\]

Answer: \( 14^{37} \)

---

#### Expression 3: \( 11^{11} \times 11^{18} \)

- Base: \( 11 \)
- Exponents: \( 11 \) and \( 18 \)

Using the Product Rule:
\[
11^{11} \times 11^{18} = 11^{11 + 18} = 11^{29}
\]

Answer: \( 11^{29} \)

---

#### Expression 4: \( 15^{10} \times 15^{17} \)

- Base: \( 15 \)
- Exponents: \( 10 \) and \( 17 \)

Using the Product Rule:
\[
15^{10} \times 15^{17} = 15^{10 + 17} = 15^{27}
\]

Answer: \( 15^{27} \)

---

#### Expression 5: \( 8^4 \times 8^6 \)

- Base: \( 8 \)
- Exponents: \( 4 \) and \( 6 \)

Using the Product Rule:
\[
8^4 \times 8^6 = 8^{4 + 6} = 8^{10}
\]

Answer: \( 8^{10} \)

---

#### Expression 6: \( 4^6 \times 4^4 \)

- Base: \( 4 \)
- Exponents: \( 6 \) and \( 4 \)

Using the Product Rule:
\[
4^6 \times 4^4 = 4^{6 + 4} = 4^{10}
\]

Answer: \( 4^{10} \)

---

#### Expression 7: \( 3^{19} \times 3^{16} \)

- Base: \( 3 \)
- Exponents: \( 19 \) and \( 16 \)

Using the Product Rule:
\[
3^{19} \times 3^{16} = 3^{19 + 16} = 3^{35}
\]

Answer: \( 3^{35} \)

---

#### Expression 8: \( 6^9 \times 6^{12} \)

- Base: \( 6 \)
- Exponents: \( 9 \) and \( 12 \)

Using the Product Rule:
\[
6^9 \times 6^{12} = 6^{9 + 12} = 6^{21}
\]

Answer: \( 6^{21} \)

---

#### Expression 9: \( 13^{20} \times 13^4 \)

- Base: \( 13 \)
- Exponents: \( 20 \) and \( 4 \)

Using the Product Rule:
\[
13^{20} \times 13^4 = 13^{20 + 4} = 13^{24}
\]

Answer: \( 13^{24} \)

---

#### Expression 10: \( 2^{16} \times 2^{11} \)

- Base: \( 2 \)
- Exponents: \( 16 \) and \( 11 \)

Using the Product Rule:
\[
2^{16} \times 2^{11} = 2^{16 + 11} = 2^{27}
\]

Answer: \( 2^{27} \)

---

#### Expression 11: \( 14^{17} \times 14^{22} \)

- Base: \( 14 \)
- Exponents: \( 17 \) and \( 22 \)

Using the Product Rule:
\[
14^{17} \times 14^{22} = 14^{17 + 22} = 14^{39}
\]

Answer: \( 14^{39} \)

---

#### Expression 12: \( 19^{16} \times 19^{12} \)

- Base: \( 19 \)
- Exponents: \( 16 \) and \( 12 \)

Using the Product Rule:
\[
19^{16} \times 19^{12} = 19^{16 + 12} = 19^{28}
\]

Answer: \( 19^{28} \)

---

#### Expression 13: \( 10^0 \times 10^1 \)

- Base: \( 10 \)
- Exponents: \( 0 \) and \( 1 \)

Using the Product Rule:
\[
10^0 \times 10^1 = 10^{0 + 1} = 10^1
\]

Answer: \( 10^1 \)

---

#### Expression 14: \( 11^2 \times 11^8 \)

- Base: \( 11 \)
- Exponents: \( 2 \) and \( 8 \)

Using the Product Rule:
\[
11^2 \times 11^8 = 11^{2 + 8} = 11^{10}
\]

Answer: \( 11^{10} \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 12^{27} \\
2. & 14^{37} \\
3. & 11^{29} \\
4. & 15^{27} \\
5. & 8^{10} \\
6. & 4^{10} \\
7. & 3^{35} \\
8. & 6^{21} \\
9. & 13^{24} \\
10. & 2^{27} \\
11. & 14^{39} \\
12. & 19^{28} \\
13. & 10^1 \\
14. & 11^{10} \\
\end{array}
}
\]