Year 8 Maths: Index Rules

Introduction to Index Rules

What Are Index Rules?

Index rules, also known as the laws of indices or exponent rules, provide a framework for handling numbers raised to powers. An index (or exponent) shows how many times a number, known as the base, is multiplied by itself. For example, $$3^4$$ means $$3 \times 3 \times 3 \times 3$$ .

Why Are Index Rules Important?

Mastering index rules allows you to simplify complex expressions, making calculations easier and quicker. It’s an essential part of mathematics in secondary school, and these rules will be applied in many future mathematical topics.

Key Index Rules

1. Multiplication Rule

When multiplying numbers with the same base, you add their indices:

$$a^m \times a^n = a^{m+n}$$

2. Division Rule

When dividing numbers with the same base, you subtract their indices:

$$\frac{a^m}{a^n} = a^{m-n}$$

3. Power of a Power Rule

When raising a power to another power, you multiply the indices:

$$(a^m)^n = a^{m \times n}$$

4. Power of One

Any number raised to the power of one is itself:

$$a^1 = a$$

5. Power of Zero

Any non-zero number raised to the power of zero is 1:

$$a^0 = 1$$

6. Negative Indices

A negative index represents a reciprocal:

$$a^{-n} = \frac{1}{a^n}$$

Index Rules Questions

Easy Level (20 Questions)

1. Simplify: $$5^2 \times 5^3$$
2. Simplify: $$4^1$$
3. Simplify: $$7^0$$
4. Write as a single power: $$6^2 \times 6^4$$
5. Simplify: $$\frac{10^5}{10^2}$$
6. Simplify: $$9^1$$
7. Simplify: $$8^3 \times 8^0$$
8. Write as a single power: $$2^3 \times 2^1$$
9. Simplify: $$\frac{5^6}{5^4}$$
10. Simplify: $$10^1$$
11. Write as a single power: $$3^5 \times 3^2$$
12. Simplify: $$\frac{6^4}{6^2}$$
13. Evaluate: $$3^0$$
14. Write as a single power: $$7^2 \times 7$$
15. Simplify: $$\frac{8^3}{8}$$
16. Evaluate: $$4^0$$
17. Simplify: $$5^2 \times 5^0$$
18. Write as a single power: $$9^3 \times 9^1$$
19. Simplify: $$2^1$$
20. Simplify: $$\frac{10^3}{10}$$

Medium Level (20 Questions)

1. Simplify: $$12^3 \times 12^2$$
2. Write as a single power: $$6^4 \times 6$$
3. Simplify: $$\frac{15^5}{15^3}$$
4. Simplify: $$8^2 \times 8^1$$
5. Write as a single power: $$11^3 \times 11^2$$
6. Simplify: $$\frac{18^4}{18^2}$$
7. Simplify: $$14^1 \times 14^2$$
8. Write as a single power: $$10^3 \times 10^1$$
9. Simplify: $$\frac{20^6}{20^4}$$
10. Write as a single power: $$4^5 \times 4^3$$
11. Simplify: $$7^2 \times 7^0 \times 7$$
12. Write as a single power: $$13^4 \times 13^2$$
13. Simplify: $$\frac{9^5}{9^2}$$
14. Write as a single power: $$16^2 \times 16$$
15. Simplify: $$\frac{21^3}{21}$$
16. Write as a single power: $$2^6 \times 2^2$$
17. Simplify: $$19^1 \times 19^0$$
18. Write as a single power: $$17^4 \times 17^3$$
19. Simplify: $$\frac{8^5}{8^3}$$
20. Write as a single power: $$15^2 \times 15$$

Hard Level (20 Questions)

1. Simplify: $$(7^2)^3$$
2. Simplify: $$(5^3)^2$$
3. Evaluate: $$4^{-2}$$
4. Simplify: $$\frac{3^5}{3^7}$$
5. Simplify: $$2^4 \times 2^{-3}$$
6. Write as a single power: $$(8^3)^2$$
7. Simplify: $$\frac{10^3}{10^5}$$
8. Write as a single power: $$6^2 \times 6^{-1}$$
9. Simplify: $$(9^2)^3$$
10. Simplify: $$(11^4)^2$$
11. Evaluate: $$5^{-3}$$
12. Simplify: $$\frac{8^7}{8^4}$$
13. Write as a single power: $$(3^5)^2$$
14. Simplify: $$10^2 \times 10^{-3}$$
15. Evaluate: $$\frac{7^4}{7^6}$$
16. Write as a single power: $$(2^3)^4$$
17. Simplify: $$(4^5)^2$$
18. Evaluate: $$9^{-1}$$
19. Simplify: $$(6^2)^3$$
20. Simplify: $$15^2 \times 15^{-4}$$

Answers and Explanations

Easy Level Answers

1. Simplify: $$5^2 \times 5^3$$ Answer: $$5^5$$ Explanation: Using the multiplication rule, add the exponents: $$2 + 3 = 5$$ .
2. Simplify: $$4^1$$ Answer: $$4$$ Explanation: Any number raised to the power of one is itself.
3. Simplify: $$7^0$$ Answer: $$1$$ Explanation: Any non-zero number raised to the power of zero is 1.
4. Write as a single power: $$6^2 \times 6^4$$ Answer: $$6^6$$ Explanation: $$2 + 4 = 6$$ .
5. Simplify: $$\frac{10^5}{10^2}$$ Answer: $$10^3$$ Explanation: Subtract the exponents: $$5 – 2 = 3$$ .
6. Simplify: $$9^1$$ Answer: $$9$$ Explanation: Any number raised to the power of one is itself.
7. Simplify: $$8^3 \times 8^0$$ Answer: $$8^3$$ Explanation: $$3 + 0 = 3$$ .
8. Write as a single power: $$2^3 \times 2^1$$ Answer: $$2^4$$ Explanation: $$3 + 1 = 4$$ .
9. Simplify: $$\frac{5^6}{5^4}$$ Answer: $$5^2$$ Explanation: $$6 – 4 = 2$$ .
10. Simplify: $$10^1$$ Answer: $$10$$ Explanation: Any number raised to the power of one is itself.
11. Write as a single power: $$3^5 \times 3^2$$ Answer: $$3^7$$ Explanation: $$5 + 2 = 7$$ .
12. Simplify: $$\frac{6^4}{6^2}$$ Answer: $$6^2$$ Explanation: $$4 – 2 = 2$$ .
13. Evaluate: $$3^0$$ Answer: $$1$$ Explanation: Any non-zero number raised to the power of zero is 1.
14. Write as a single power: $$7^2 \times 7$$ Answer: $$7^3$$ Explanation: $$2 + 1 = 3$$ .
15. Simplify: $$\frac{8^3}{8}$$ Answer: $$8^2$$ Explanation: $$3 – 1 = 2$$ .
16. Evaluate: $$4^0$$ Answer: $$1$$ Explanation: Any non-zero number raised to the power of zero is 1.
17. Simplify: $$5^2 \times 5^0$$ Answer: $$5^2$$ Explanation: $$2 + 0 = 2$$ .
18. Write as a single power: $$9^3 \times 9^1$$ Answer: $$9^4$$ Explanation: $$3 + 1 = 4$$ .
19. Simplify: $$2^1$$ Answer: $$2$$ Explanation: Any number raised to the power of one is itself.
20. Simplify: $$\frac{10^3}{10}$$ Answer: $$10^2$$ Explanation: $$3 – 1 = 2$$ .

Medium Level Answers

1. Simplify: $$12^3 \times 12^2$$ Answer: $$12^5$$ Explanation: $$3 + 2 = 5$$ .
2. Write as a single power: $$6^4 \times 6$$ Answer: $$6^5$$ Explanation: $$4 + 1 = 5$$ .
3. Simplify: $$\frac{15^5}{15^3}$$ Answer: $$15^2$$ Explanation: $$5 – 3 = 2$$ .
4. Simplify: $$8^2 \times 8^1$$ Answer: $$8^3$$ Explanation: $$2 + 1 = 3$$ .
5. Write as a single power: $$11^3 \times 11^2$$ Answer: $$11^5$$ Explanation: $$3 + 2 = 5$$ .
6. Simplify: $$\frac{18^4}{18^2}$$ Answer: $$18^2$$ Explanation: $$4 – 2 = 2$$ .
7. Simplify: $$14^1 \times 14^2$$ Answer: $$14^3$$ Explanation: $$1 + 2 = 3$$ .
8. Write as a single power: $$10^3 \times 10^1$$ Answer: $$10^4$$ Explanation: $$3 + 1 = 4$$ .
9. Simplify: $$\frac{20^6}{20^4}$$ Answer: $$20^2$$ Explanation: $$6 – 4 = 2$$ .
10. Write as a single power: $$4^5 \times 4^3$$ Answer: $$4^8$$ Explanation: $$5 + 3 = 8$$ .
11. Simplify: $$7^2 \times 7^0 \times 7$$ Answer: $$7^3$$ Explanation: $$2 + 0 + 1 = 3$$ .
12. Write as a single power: $$13^4 \times 13^2$$ Answer: $$13^6$$ Explanation: $$4 + 2 = 6$$ .
13. Simplify: $$\frac{9^5}{9^2}$$ Answer: $$9^3$$ Explanation: $$5 – 2 = 3$$ .
14. Write as a single power: $$16^2 \times 16$$ Answer: $$16^3$$ Explanation: $$2 + 1 = 3$$ .
15. Simplify: $$\frac{21^3}{21}$$ Answer: $$21^2$$ Explanation: $$3 – 1 = 2$$ .
16. Write as a single power: $$2^6 \times 2^2$$ Answer: $$2^8$$ Explanation: $$6 + 2 = 8$$ .
17. Simplify: $$19^1 \times 19^0$$ Answer: $$19^1$$ Explanation: $$1 + 0 = 1$$ .
18. Write as a single power: $$17^4 \times 17^3$$ Answer: $$17^7$$ Explanation: $$4 + 3 = 7$$ .
19. Simplify: $$\frac{8^5}{8^3}$$ Answer: $$8^2$$ Explanation: $$5 – 3 = 2$$ .
20. Write as a single power: $$15^2 \times 15$$ Answer: $$15^3$$ Explanation: $$2 + 1 = 3$$ .

Hard Level Answers

1. Simplify: $$(7^2)^3$$ Answer: $$7^6$$ Explanation: Multiply the exponents: $$2 \times 3 = 6$$ .
2. Simplify: $$(5^3)^2$$ Answer: $$5^6$$ Explanation: $$3 \times 2 = 6$$ .
3. Evaluate: $$4^{-2}$$ Answer: $$\frac{1}{4^2} = \frac{1}{16}$$ Explanation: Negative exponent indicates reciprocal.
4. Simplify: $$\frac{3^5}{3^7}$$ Answer: $$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$ Explanation: $$5 – 7 = -2$$ .
5. Simplify: $$2^4 \times 2^{-3}$$ Answer: $$2^{1} = 2$$ Explanation: $$4 + (-3) = 1$$ .
6. Write as a single power: $$(8^3)^2$$ Answer: $$8^6$$ Explanation: $$3 \times 2 = 6$$ .
7. Simplify: $$\frac{10^3}{10^5}$$ Answer: $$10^{-2} = \frac{1}{10^2} = \frac{1}{100}$$ Explanation: $$3 – 5 = -2$$ .
8. Write as a single power: $$6^2 \times 6^{-1}$$ Answer: $$6^{1} = 6$$ Explanation: $$2 + (-1) = 1$$ .
9. Simplify: $$(9^2)^3$$ Answer: $$9^6$$ Explanation: $$2 \times 3 = 6$$ .
10. Simplify: $$(11^4)^2$$ Answer: $$11^8$$ Explanation: $$4 \times 2 = 8$$ .
11. Evaluate: $$5^{-3}$$ Answer: $$\frac{1}{5^3} = \frac{1}{125}$$ Explanation: Negative exponent indicates reciprocal.
12. Simplify: $$\frac{8^7}{8^4}$$ Answer: $$8^3$$ Explanation: $$7 – 4 = 3$$ .
13. Write as a single power: $$(3^5)^2$$ Answer: $$3^{10}$$ Explanation: $$5 \times 2 = 10$$ .
14. Simplify: $$10^2 \times 10^{-3}$$ Answer: $$10^{-1} = \frac{1}{10}$$ Explanation: $$2 + (-3) = -1$$ .
15. Evaluate: $$\frac{7^4}{7^6}$$ Answer: $$7^{-2} = \frac{1}{7^2} = \frac{1}{49}$$ Explanation: $$4 – 6 = -2$$ .
16. Write as a single power: $$(2^3)^4$$ Answer: $$2^{12}$$ Explanation: $$3 \times 4 = 12$$ .
17. Simplify: $$(4^5)^2$$ Answer: $$4^{10}$$ Explanation: $$5 \times 2 = 10$$ .
18. Evaluate: $$9^{-1}$$ Answer: $$\frac{1}{9}$$ Explanation: Negative exponent indicates reciprocal.
19. Simplify: $$(6^2)^3$$ Answer: $$6^6$$ Explanation: $$2 \times 3 = 6$$ .
20. Simplify: $$15^2 \times 15^{-4}$$ Answer: $$15^{-2} = \frac{1}{15^2} = \frac{1}{225}$$ Explanation: $$2 + (-4) = -2$$ .

Summary of Index Rules

Understanding and applying index rules is crucial for simplifying expressions involving exponents. Remember the key rules:

• Multiplication: Add the exponents when multiplying like bases.
• Division: Subtract the exponents when dividing like bases.
• Power of a Power: Multiply the exponents.
• Power of One: Any base to the power of one is the base itself.
• Power of Zero: Any non-zero base to the power of zero is one.
• Negative Indices: A negative exponent indicates the reciprocal of the base raised to the positive exponent.

Practicing these rules through various levels of questions will enhance your mathematical proficiency and prepare you for more advanced topics.