Index notation-multiplication and division laws worksheet (with ... - Free Printable

It looks like you've uploaded a page from a math worksheet on indices (exponents), focusing on the rules of multiplying and dividing powers with the same base. The task is to simplify expressions by applying the laws of indices.

Let’s go through the rules first, then solve each problem step-by-step.

---

🔷 Rules of Indices (Powers):

1. Multiplying Powers:
$ a^m \times a^n = a^{m+n} $

2. Dividing Powers:
$ \frac{a^m}{a^n} = a^{m-n} $

3. Power of a Power:
$ (a^m)^n = a^{m \times n} $

4. Zero Power:
$ a^0 = 1 $ (for $ a \neq 0 $)

5. Negative Power:
$ a^{-n} = \frac{1}{a^n} $

6. Power of a Product:
$ (ab)^n = a^n b^n $

---

Now, let's solve each question in the Exercise section:

---

✔ Exercise: Write the following as a single power

#### 1. $ y^5 \times y^3 $
- Same base $ y $, so add exponents:
- $ y^{5+3} = y^8 $

✔ Answer: $ y^8 $

---

#### 2. $ 2^7 \times 2^4 $
- Same base $ 2 $, add exponents:
- $ 2^{7+4} = 2^{11} $

✔ Answer: $ 2^{11} $

---

#### 3. $ 8^3 \times 8^2 $
- Same base $ 8 $, add exponents:
- $ 8^{3+2} = 8^5 $

✔ Answer: $ 8^5 $

---

#### 4. $ 6^9 \div 6^3 $
- Same base $ 6 $, subtract exponents:
- $ 6^{9-3} = 6^6 $

✔ Answer: $ 6^6 $

---

#### 5. $ 12^4 \times 12^5 $
- Same base $ 12 $, add exponents:
- $ 12^{4+5} = 12^9 $

✔ Answer: $ 12^9 $

---

#### 6. $ 10 \times 10^2 \times 10^3 $
- Note: $ 10 = 10^1 $
- So: $ 10^1 \times 10^2 \times 10^3 = 10^{1+2+3} = 10^6 $

✔ Answer: $ 10^6 $

---

#### 7. $ y^3 \times y^5 $
- Same base $ y $, add exponents:
- $ y^{3+5} = y^8 $

✔ Answer: $ y^8 $

---

#### 8. $ x^4 \div x^3 $
- Same base $ x $, subtract exponents:
- $ x^{4-3} = x^1 = x $

✔ Answer: $ x $

---

#### 9. $ 8^3 \times 8^2 $
- Already done in #3 → $ 8^{5} $

✔ Answer: $ 8^5 $

---

#### 10. $ a^3 \times a^5 $
- Add exponents: $ a^{3+5} = a^8 $

✔ Answer: $ a^8 $

---

#### 11. $ 2^{10} \div 2^4 $
- Subtract exponents: $ 2^{10-4} = 2^6 $

✔ Answer: $ 2^6 $

---

#### 12. $ 3^7 \div 3^5 $
- $ 3^{7-5} = 3^2 $

✔ Answer: $ 3^2 $

---

#### 13. $ x^3 \times x^2 $
- $ x^{3+2} = x^5 $

✔ Answer: $ x^5 $

---

#### 14. $ x^4 \times x^2 $
- $ x^{4+2} = x^6 $

✔ Answer: $ x^6 $

---

#### 15. $ y^5 \div y^2 $
- $ y^{5-2} = y^3 $

✔ Answer: $ y^3 $

---

#### 16. $ b^7 \div b^3 $
- $ b^{7-3} = b^4 $

✔ Answer: $ b^4 $

---

#### 17. $ y^3 \times y^2 \times y $
- Note: $ y = y^1 $
- $ y^{3+2+1} = y^6 $

✔ Answer: $ y^6 $

---

#### 18. $ t \times t^2 \times t^3 $
- $ t = t^1 $
- $ t^{1+2+3} = t^6 $

✔ Answer: $ t^6 $

---

#### 19. $ x^4 \div x^2 $
- $ x^{4-2} = x^2 $

✔ Answer: $ x^2 $

---

#### 20. $ b^7 \times b^3 \times b^2 $
- $ b^{7+3+2} = b^{12} $

✔ Answer: $ b^{12} $

---

✔ Final Answers Summary:

| Q | Answer |
|---|--------|
| 1 | $ y^8 $ |
| 2 | $ 2^{11} $ |
| 3 | $ 8^5 $ |
| 4 | $ 6^6 $ |
| 5 | $ 12^9 $ |
| 6 | $ 10^6 $ |
| 7 | $ y^8 $ |
| 8 | $ x $ |
| 9 | $ 8^5 $ |
|10 | $ a^8 $ |
|11 | $ 2^6 $ |
|12 | $ 3^2 $ |
|13 | $ x^5 $ |
|14 | $ x^6 $ |
|15 | $ y^3 $ |
|16 | $ b^4 $ |
|17 | $ y^6 $ |
|18 | $ t^6 $ |
|19 | $ x^2 $ |
|20 | $ b^{12} $ |

---

Let me know if you'd like this formatted as a printable PDF or need help with more advanced index problems!