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.