Year 8 Maths: Indices

Indices, also known as powers or exponents, are a way of expressing repeated multiplication of the same number. They simplify expressions that involve multiplying a number by itself multiple times.

For example, instead of writing:

$$2 \times 2 \times 2 \times 2 = 16$$

We can write this more simply using indices:

$$2^4 = 16$$

Here, the number 2 is called the base and the number 4 is the index or power. The index tells you how many times to multiply the base by itself.

Key Concepts in Indices

1. Multiplying with the Same Base:

When multiplying two numbers with the same base, you add the indices.

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

Example:
$$2^3 \times 2^2 = 2^{3+2} = 2^5$$

2. Dividing with the Same Base:

When dividing two numbers with the same base, you subtract the indices.

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

Example:
$$\frac{3^5}{3^2} = 3^{5-2} = 3^3$$

3. Raising a Power to Another Power:

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

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

Example:
$$(4^2)^3 = 4^{2 \times 3} = 4^6$$

4. Zero as an Index:

Any non-zero number raised to the power of 0 is always 1.

$$a^0 = 1$$

Example:
$$5^0 = 1$$

5. Negative Indices:

A negative index indicates a reciprocal.

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

Example:
$$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$

Practice Questions on Indices

Easy Level

1. Simplify $$2^3$$
2. Simplify $$5^2$$
3. Simplify $$4^3$$
4. Simplify $$10^2$$
5. Simplify $$3^3$$
6. What is $$7^0$$ ?
7. Simplify $$6^2$$
8. What is $$9^0$$ ?
9. Simplify $$2^4$$
10. What is $$8^0$$ ?
11. Simplify $$2^1$$
12. What is $$3^1$$ ?
13. Simplify $$1^4$$
14. What is $$5^0$$ ?
15. Simplify $$2 \times 2^3$$
16. Simplify $$4^2$$
17. What is $$10^0$$ ?
18. Simplify $$3^2$$
19. Simplify $$2^5$$
20. Simplify $$7^1$$

Medium Level

1. Simplify $$2^3 \times 2^2$$
2. Simplify $$3^4 \div 3^2$$
3. Simplify $$(5^2)^3$$
4. Simplify $$4^3 \times 4^2$$
5. Simplify $$\frac{6^5}{6^2}$$
6. What is $$(2^4)^2$$ ?
7. Simplify $$3^2 \times 3^3$$
8. Simplify $$\frac{9^4}{9^3}$$
9. What is $$(5^3)^2$$ ?
10. Simplify $$7^4 \times 7^1$$
11. Simplify $$\frac{8^5}{8^3}$$
12. Simplify $$(4^2)^3$$
13. Simplify $$2^3 \times 2^4$$
14. Simplify $$\frac{10^5}{10^3}$$
15. Simplify $$(3^3)^2$$
16. Simplify $$6^4 \times 6^2$$
17. Simplify $$\frac{7^6}{7^4}$$
18. Simplify $$(2^5)^2$$
19. Simplify $$9^3 \div 9^2$$
20. Simplify $$(5^2)^4$$

Hard Level

1. Simplify $$(3^2)^4$$
2. Simplify $$5^3 \times 5^4$$
3. Simplify $$\frac{4^6}{4^2}$$
4. Simplify $$(2^3)^5$$
5. Simplify $$\frac{9^6}{9^3}$$
6. Simplify $$(5^4)^2$$
7. Simplify $$8^6 \div 8^4$$
8. Simplify $$7^5 \times 7^2$$
9. Simplify $$(6^3)^4$$
10. Simplify $$3^4 \times 3^5$$
11. Simplify $$(2^5)^3$$
12. Simplify $$\frac{10^7}{10^4}$$
13. Simplify $$(4^4)^2$$
14. Simplify $$\frac{7^8}{7^3}$$
15. Simplify $$(3^6)^2$$
16. Simplify $$9^4 \div 9^2$$
17. Simplify $$\frac{12^6}{12^2}$$
18. Simplify $$11^3 \times 11^2$$
19. Simplify $$(5^3)^4$$
20. Simplify $$\frac{10^8}{10^5}$$

Answers and Explanations

Easy Level

1. $$2^3 = 8$$

• $$2 \times 2 \times 2 = 8$$

1. $$5^2 = 25$$

• $$5 \times 5 = 25$$

1. $$4^3 = 64$$

• $$4 \times 4 \times 4 = 64$$

1. $$10^2 = 100$$

• $$10 \times 10 = 100$$

1. $$3^3 = 27$$

• $$3 \times 3 \times 3 = 27$$

1. $$7^0 = 1$$

• Any number raised to the power of 0 is 1.

1. $$6^2 = 36$$

• $$6 \times 6 = 36$$

1. $$9^0 = 1$$

• Any number raised to the power of 0 is 1.

1. $$2^4 = 16$$

• $$2 \times 2 \times 2 \times 2 = 16$$

1. $$8^0 = 1$$
   • Any number raised to the power of 0 is 1.
2. $$2^1 = 2$$
   • Any number raised to the power of 1 is itself.
3. $$3^1 = 3$$
4. $$1^4 = 1$$
   • Any power of 1 is 1.
5. $$5^0 = 1$$
6. $$2 \times 2^3 = 2^4 = 16$$
7. $$4^2 = 16$$
8. $$10^0 = 1$$
9. $$3^2 = 9$$
10. $$2^5 = 32$$
11. $$7^1 = 7$$

Medium Level

1. $$2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$$
2. $$3^4 \div 3^2 = 3^{4-2} = 3^2 = 9$$
3. $$(5^2)^3 = 5^{2 \times 3} = 5^6 = 15625$$
4. $$4^3 \times 4^2 = 4^{3+2} = 4^5 = 1024$$
5. $$\frac{6^5}{6^2} = 6^{5-2} = 6^3 = 216$$
6. $$(2^4)^2 = 2^{4 \times 2} = 2^8 = 256$$
7. $$3^2 \times 3^3 = 3^{2+3} = 3^5 = 243$$
8. $$ \frac{9^4}{9^3} = 9^{4-3} = 9^

1 = 9 $$

9. $$(5^3)^2 = 5^{3 \times 2} = 5^6 = 15625$$
10. $$7^4 \times 7^1 = 7^{4+1} = 7^5 = 16807$$
11. $$\frac{8^5}{8^3} = 8^{5-3} = 8^2 = 64$$
12. $$(4^2)^3 = 4^{2 \times 3} = 4^6 = 4096$$
13. $$2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$$
14. $$\frac{10^5}{10^3} = 10^{5-3} = 10^2 = 100$$
15. $$(3^3)^2 = 3^{3 \times 2} = 3^6 = 729$$
16. $$6^4 \times 6^2 = 6^{4+2} = 6^6 = 46656$$
17. $$\frac{7^6}{7^4} = 7^{6-4} = 7^2 = 49$$
18. $$(2^5)^2 = 2^{5 \times 2} = 2^{10} = 1024$$
19. $$9^3 \div 9^2 = 9^{3-2} = 9^1 = 9$$
20. $$(5^2)^4 = 5^{2 \times 4} = 5^8 = 390625$$

Hard Level

1. $$(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$$
2. $$5^3 \times 5^4 = 5^{3+4} = 5^7 = 78125$$
3. $$\frac{4^6}{4^2} = 4^{6-2} = 4^4 = 256$$
4. $$(2^3)^5 = 2^{3 \times 5} = 2^{15} = 32768$$
5. $$\frac{9^6}{9^3} = 9^{6-3} = 9^3 = 729$$
6. $$(5^4)^2 = 5^{4 \times 2} = 5^8 = 390625$$
7. $$8^6 \div 8^4 = 8^{6-4} = 8^2 = 64$$
8. $$7^5 \times 7^2 = 7^{5+2} = 7^7 = 823543$$
9. $$(6^3)^4 = 6^{3 \times 4} = 6^{12} = 2176782336$$
10. $$3^4 \times 3^5 = 3^{4+5} = 3^9 = 19683$$
11. $$(2^5)^3 = 2^{5 \times 3} = 2^{15} = 32768$$
12. $$\frac{10^7}{10^4} = 10^{7-4} = 10^3 = 1000$$
13. $$(4^4)^2 = 4^{4 \times 2} = 4^8 = 65536$$
14. $$\frac{7^8}{7^3} = 7^{8-3} = 7^5 = 16807$$
15. $$(3^6)^2 = 3^{6 \times 2} = 3^{12} = 531441$$
16. $$9^4 \div 9^2 = 9^{4-2} = 9^2 = 81$$
17. $$\frac{12^6}{12^2} = 12^{6-2} = 12^4 = 20736$$
18. $$11^3 \times 11^2 = 11^{3+2} = 11^5 = 161051$$
19. $$(5^3)^4 = 5^{3 \times 4} = 5^{12} = 244140625$$
20. $$\frac{10^8}{10^5} = 10^{8-5} = 10^3 = 1000$$

This set of questions and answers covers a range of difficulties related to indices, helping students to understand and practice the key rules such as multiplication, division, and raising powers.