Exponents & Powers (Laws of Exponents) worksheet - Free Printable

Sure! Let’s solve each problem using the Laws of Exponents. Here are the key rules we’ll use:

---

🔢 Laws of Exponents:

1. Product of Powers (Same Base):
\( a^m \times a^n = a^{m+n} \)

2. Quotient of Powers (Same Base):
\( a^m \div a^n = a^{m-n} \)

3. Power of a Product:
\( a^m \times b^m = (a \times b)^m \) — *only if exponents are same*

4. Combining operations:
Follow order of operations (BODMAS/PEMDAS), simplify inside parentheses first.

---

Now, let’s solve each problem step by step:

---

✔ 1. \( 3^2 \times 3^4 \times 3^8 \)

→ All bases are same (3), so add exponents:
\( 3^{2+4+8} = 3^{14} \)

Answer: \( \boxed{3^{14}} \)

---

✔ 2. \( 6^{15} \div 6^{10} \)

→ Same base, subtract exponents:
\( 6^{15-10} = 6^5 \)

Answer: \( \boxed{6^5} \)

---

✔ 3. \( a^3 \times a^2 \)

→ Same base ‘a’, add exponents:
\( a^{3+2} = a^5 \)

Answer: \( \boxed{a^5} \)

---

✔ 4. \( 7^3 \times 7^2 \)

→ Same base, add exponents:
\( 7^{3+2} = 7^5 \)

Answer: \( \boxed{7^5} \)

---

✔ 5. \( 2^5 \times 5^5 \)

→ Different bases, but same exponent → Use Power of a Product:
\( (2 \times 5)^5 = 10^5 \)

Answer: \( \boxed{10^5} \)

---

✔ 6. \( a^4 \times b^4 \)

→ Different bases, same exponent →
\( (a \times b)^4 = (ab)^4 \)

Answer: \( \boxed{(ab)^4} \)

---

✔ 7. \( (2^{20} \div 2^{15}) \times 2^3 \)

→ First, simplify inside parentheses:
\( 2^{20-15} = 2^5 \)
Then multiply by \( 2^3 \):
\( 2^5 \times 2^3 = 2^{5+3} = 2^8 \)

Answer: \( \boxed{2^8} \)

---

✔ 8. \( 8^8 \div 8^2 \)

→ Same base, subtract exponents:
\( 8^{8-2} = 8^6 \)

Answer: \( \boxed{8^6} \)

---

✔ 9. \( 3^4 \times 3^3 \times 3^6 \)

→ Same base, add exponents:
\( 3^{4+3+6} = 3^{13} \)

Answer: \( \boxed{3^{13}} \)

---

✔ 10. \( 6^{15} \times 6^{10} \)

→ Same base, add exponents:
\( 6^{15+10} = 6^{25} \)

Answer: \( \boxed{6^{25}} \)

---

## 🎉 Final Answers Summary:

1. \( \boxed{3^{14}} \)
2. \( \boxed{6^5} \)
3. \( \boxed{a^5} \)
4. \( \boxed{7^5} \)
5. \( \boxed{10^5} \)
6. \( \boxed{(ab)^4} \)
7. \( \boxed{2^8} \)
8. \( \boxed{8^6} \)
9. \( \boxed{3^{13}} \)
10. \( \boxed{6^{25}} \)

---

Let me know if you’d like these explained with diagrams or examples for a younger student! 😊