🎬 See repeated multiplication
A power is a shorter way to show repeated multiplication.
🎯 By the end of this lesson...
You should be able to:
- Explain what a power represents.
- Identify the base and index in a power.
- Calculate squares and cubes.
- Recognise common square and cube numbers.
- Find square roots of perfect squares.
- Find cube roots of perfect cubes.
- Distinguish between a number and its square or cube.
- Use powers and roots in practical situations.
🤔 Think About This...
Building a square patio
A builder has exactly 64 identical tiles.
The tiles must form a square, with the same number of tiles along each side.
How many tiles should be placed along each side?
We are looking for a number that gives 64 when multiplied by itself.
We can write this as √64 = 8.
1️⃣ What Is a Power?
A power is a short way of writing repeated multiplication.
The number being multiplied.
How many times the base is used as a factor.
⚠ What a Power Does Not Mean
❌ Incorrect
✅ Correct
🧠 StudyNest Method: Evaluating a Power
Step 2: Identify the index.
Step 3: Write the base as a factor the number of times shown by the index.
Step 4: Multiply carefully.
Step 5: Check that you have used the correct number of factors.
Example 1: Evaluate 3⁴
💡 Did You Know?
Powers of 2 are important in computers because computers process information using two basic states.
This is one reason powers appear in computing, storage and digital technology.
2️⃣ Square Numbers
To square a number, multiply it by itself.
The expression 4² is read as:
3️⃣ Perfect Squares
A perfect square is the result of multiplying a whole number by itself.
🧠 Spot the Pattern
Look at the differences between consecutive square numbers.
3, 5, 7, 9, ...
4️⃣ Square Roots
A square root reverses the process of squaring.
The symbol for a square root is:
“What positive number multiplied by itself gives 36?”
🧠 StudyNest Method: Finding a Square Root
Step 2: Ask which positive number gives that result when multiplied by itself.
Step 3: Check by squaring your answer.
Example 2: Find √81
Ask:
Example 3: Find √144
⚠ Square Roots and Irrational Numbers
Not every square root gives a whole number.
Perfect square
49 is a perfect square, so its square root is a whole number.
Not a perfect square
The decimal does not end or repeat, so √50 is irrational.
Simplify or evaluate it first.
🌍 Real-Life Example: Tree Nursery
Example 4
A nursery has 196 young trees.
The trees must be arranged in a square, with the same number of trees in each row and column.
Find the number of trees in each row.
5️⃣ Cube Numbers
To cube a number, multiply it by itself three times.
The expression 3³ is read as:
6️⃣ Perfect Cubes
A perfect cube is the result of multiplying a whole number by itself three times.
7️⃣ Cube Roots
A cube root reverses the process of cubing.
“What number multiplied by itself three times gives 64?”
🧠 StudyNest Method: Finding a Cube Root
Step 2: Ask which number gives that result when multiplied by itself three times.
Step 3: Check by cubing your answer.
Example 5: Find ∛125
Example 6: Find ∛216
🌍 Real-Life Example: Packing Small Boxes
Example 7
A storage container holds 125 identical small boxes.
The boxes are arranged equally in length, width and height.
How many boxes are arranged along each direction?
8️⃣ Comparing Squares and Cubes
Square
The base is used twice as a factor.
Cube
The base is used three times as a factor.
The index tells you how many factors are required.
9️⃣ Powers and Directed Numbers
Brackets are important when powers involve negative numbers.
Example 8: Evaluate (−3)²
Example 9: Evaluate (−3)³
Cubing a negative number gives a negative result.
🔟 Powers in the Order of Operations
Powers and roots are completed before multiplication, division, addition and subtraction.
Example 10: Calculate 5 + 2 × 3²
Calculate the power first:
Multiply:
🧠 The StudyNest Powers and Roots Strategy
power, square root or cube root.
Step 2: If it is a power, identify the base and index.
Step 3: If it is a root, ask what number produced the value.
Step 4: Write repeated multiplication where helpful.
Step 5: Calculate carefully.
Step 6: Check by reversing the operation.
🎯 Exam Tips
⚠ Common Mistakes
Thinking 5² means 5 × 2.
5² means 5 × 5.
Thinking 3³ = 9.
3³ = 3 × 3 × 3 = 27.
Dividing a number by 2 when finding its square root.
√36 is 6, not 18.
Assuming every square root is irrational.
Square roots of perfect squares are rational.
Forgetting that a cube requires three equal factors.
Ignoring brackets when raising a negative number to a power.
🎮 Your Turn!
🟢 Foundation Questions
- State the base and index in 7³.
- Write 4³ as repeated multiplication.
- Calculate 6².
- Calculate 5³.
- Find √49.
- Find √121.
- Find ∛64.
- Find ∛1000.
🔵 Developing Questions
- State whether 81 is a perfect square.
- State whether 50 is a perfect square.
- Evaluate 2⁵.
- Evaluate (−4)².
- Evaluate (−4)³.
- Calculate 10 + 3² × 2.
- Calculate 40 − √25 × 4.
🔴 Challenge Questions
- A square garden has an area of 225 m². Find the length of one side.
- 343 small cubes are arranged to form one large cube. How many small cubes are arranged along each edge?
- Find the two consecutive whole numbers between which √70 lies.
- A square classroom floor contains 324 equal square tiles. Find the number of tiles along one side.
- Calculate 2 × [3² + ∛64].
✅ Show Answers
- Base = 7, index = 3
- 4 × 4 × 4
- 36
- 125
- 7
- 11
- 4
- 10
- Yes, because 9² = 81.
- No. It lies between 7² = 49 and 8² = 64.
- 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
- (−4)² = 16
- (−4)³ = −64
-
10 + 9 × 2
= 10 + 18
= 28 -
40 − 5 × 4
= 40 − 20
= 20 - √225 = 15 m
- ∛343 = 7 cubes
-
8² = 64 and 9² = 81
Therefore, √70 lies between 8 and 9. - √324 = 18 tiles
-
2 × [9 + 4]
= 2 × 13
= 26
🧠 Remember This
n² means n × n.
n³ means n × n × n.
A square root reverses squaring.
A cube root reverses cubing.
Perfect squares and cubes give whole-number roots.
Always check roots by reversing the operation.
🎯 Before Moving On...
- ✅ Can I identify the base and index?
- ✅ Can I write a power as repeated multiplication?
- ✅ Can I calculate squares and cubes?
- ✅ Can I recognise common perfect squares?
- ✅ Can I recognise common perfect cubes?
- ✅ Can I calculate square roots?
- ✅ Can I calculate cube roots?
- ✅ Can I work with powers of negative numbers?
- ✅ Can I apply powers and roots in practical problems?
🎉 Well Done!
You can now calculate powers, squares, cubes and roots and explain what each operation means.
Read the symbol → Identify what it means → Write the multiplication or reverse operation → Check the answer
📈 Your Progress
Lesson 9 of 17