Currently Empty: $0.00
Power and Roots
Year 8 • Number
Lesson: Powers and Roots
Learn how to work with powers, squares, cubes and roots with clear explanations, step-by-step examples, and practice questions – ideal for Year 8 students.
Lesson Objectives
- Understand what powers (indices) are and how to read them.
- Calculate squares and cubes of whole numbers.
- Find simple square roots and cube roots.
- Use basic rules of powers to simplify expressions.
1. What Are Powers?
A power (or index/exponent) is a way of writing repeated multiplication.
Example: 34 means 3 × 3 × 3 × 3.
- Base: 3
- Index (or exponent): 4
- Value: 34 = 81
Quick examples:
23 = 2 × 2 × 2 = 8
52 = 5 × 5 = 25
101 = 10
23 = 2 × 2 × 2 = 8
52 = 5 × 5 = 25
101 = 10
2. Squares and Cubes
A square of a number is that number raised to the power 2.
A cube of a number is that number raised to the power 3.
Squares (n2)
22 = 4
32 = 9
42 = 16
52 = 25
102 = 100
22 = 4
32 = 9
42 = 16
52 = 25
102 = 100
Cubes (n3)
23 = 8
33 = 27
43 = 64
53 = 125
23 = 8
33 = 27
43 = 64
53 = 125
3. Square Roots and Cube Roots
A square root of a number is a value that, when multiplied by itself, gives the original number.
A cube root of a number is a value that, when multiplied by itself three times, gives the original number.
Examples:
√25 = 5 because 5 × 5 = 25
√81 = 9 because 9 × 9 = 81
∛27 = 3 because 3 × 3 × 3 = 27
∛64 = 4 because 4 × 4 × 4 = 64
√25 = 5 because 5 × 5 = 25
√81 = 9 because 9 × 9 = 81
∛27 = 3 because 3 × 3 × 3 = 27
∛64 = 4 because 4 × 4 × 4 = 64
4. Basic Rules of Powers
For Year 8, we focus on simple rules with the same base:
- Multiplying powers: am × an = am+n
- Dividing powers: am ÷ an = am−n (m > n)
- Power of a power: (am)n = am×n
Examples:
23 × 24 = 27 = 128
54 ÷ 52 = 52 = 25
(32)3 = 36 = 729
23 × 24 = 27 = 128
54 ÷ 52 = 52 = 25
(32)3 = 36 = 729
5. Real-Life Example
A small cube has side length 3 cm. A larger cube has side length 5 cm.
Which cube has the greater volume, and by how much?
Volume of a cube = side3
Small cube: 33 = 27 cm³
Large cube: 53 = 125 cm³
Difference: 125 − 27 = 98 cm³
Practice Questions
A. Write as a power or in expanded form
- Write in expanded form: 43
- Write as a power of 2: 2 × 2 × 2 × 2
- Write as a power of 5: 5 × 5 × 5
B. Evaluate the powers
- 32
- 62
- 24
- 53
C. Squares and Roots
- Find: √49
- Find: √121
- Find: ∛27
- Find: ∛125
D. Use Rules of Powers
- 23 × 22 = ?
- 54 ÷ 52 = ?
- (22)3 = ?
- 35 ÷ 32 = ?
E. Word Problem
-
A square field has side length 12 m. What is the area of the field?
(Hint: Area of square = side2) -
A cube-shaped box has side length 4 cm. What is its volume?
(Hint: Volume of cube = side3)
✅ Show Answer Key
A. Write as a power / expanded form
- 43 = 4 × 4 × 4
- 2 × 2 × 2 × 2 = 24
- 5 × 5 × 5 = 53
B. Evaluate the powers
- 32 = 9
- 62 = 36
- 24 = 16
- 53 = 125
C. Squares and Roots
- √49 = 7
- √121 = 11
- ∛27 = 3
- ∛125 = 5
D. Use Rules of Powers
- 23 × 22 = 25 = 32
- 54 ÷ 52 = 52 = 25
- (22)3 = 26 = 64
- 35 ÷ 32 = 33 = 27
E. Word Problems
- Area = 122 = 144 m²
- Volume = 43 = 64 cm³
© Aviate Learning – Powers & Roots (Year 8)