N6: Powers and Roots
Use positive integer powers and roots; square, cube and higher; recognise powers of 2, 3, 4, 5; estimate powers and roots
📋 Key Concepts
Powers: A power tells you how many times to multiply a number by itself. 5² means 5 × 5 = 25.
Roots: A root is the inverse of a power. If 5² = 25, then √25 = 5.
Key notation:
x² = x squared (x × x)
x³ = x cubed (x × x × x)
√x = square root of x
³√x = cube root of x
x² = x squared (x × x)
x³ = x cubed (x × x × x)
√x = square root of x
³√x = cube root of x
📝 Square Numbers
Square number: The result of multiplying a number by itself. You should memorise squares up to 15².
| Number | Square | Number | Square |
|---|---|---|---|
| 1² | 1 | 9² | 81 |
| 2² | 4 | 10² | 100 |
| 3² | 9 | 11² | 121 |
| 4² | 16 | 12² | 144 |
| 5² | 25 | 13² | 169 |
| 6² | 36 | 14² | 196 |
| 7² | 49 | 15² | 225 |
| 8² | 64 | 20² | 400 |
📝 Cube Numbers
Cube number: The result of multiplying a number by itself twice. Memorise cubes up to 5³.
| Number | Cube |
|---|---|
| 1³ | 1 |
| 2³ | 8 |
| 3³ | 27 |
| 4³ | 64 |
| 5³ | 125 |
| 10³ | 1000 |
📝 Square Roots
Square root: Finding what number, when multiplied by itself, gives the original number.
Example 1
Find: √144
Solution: What number squared gives 144?
12² = 144, so √144 = 12
Example 2
Find: √0.64
Solution: 0.8² = 0.64, so √0.64 = 0.8
📝 Estimating Roots
Method: Find the two square numbers either side, then estimate between them.
Example 3
Estimate √20
Solution:
16 < 20 < 25
4² = 16 and 5² = 25
√20 is between 4 and 5, closer to 5
Estimate: 4.5 (actual: 4.47...)
Example 4
Estimate ³√30
Solution:
27 < 30 < 64
3³ = 27 and 4³ = 64
³√30 is between 3 and 4, very close to 3
Estimate: 3.1 (actual: 3.11...)
📝 Higher Powers
Powers of 2, 3, 4, 5: You should recognise these common powers.
| Power | Value |
|---|---|
| 2⁴ | 16 |
| 2⁵ | 32 |
| 2⁶ | 64 |
| 3⁴ | 81 |
| 4³ | 64 |
| 5³ | 125 |
Example 5
Calculate: 2⁴ + 3³
Solution:
2⁴ = 2 × 2 × 2 × 2 = 16
3³ = 3 × 3 × 3 = 27
16 + 27 = 43
❓ Practice Questions
Q1: Calculate: 7² and 11²
Q2: Calculate: √81 and √196
Q3: Calculate: 6³ and ³√125
Q4: Estimate √50 to one decimal place
Q5: Calculate: 2⁵ - 3⁴
✅ Answers
- 7² = 49, 11² = 121
- √81 = 9, √196 = 14
- 6³ = 216, ³√125 = 5
- 49 < 50 < 64, √50 ≈ 7.1 (actual: 7.07)
- 32 - 81 = -49
🎯 Exam Tips
- Memorise squares up to 15² and cubes up to 5³
- When estimating roots, always state the two perfect squares/cubes either side first
- Remember: √100 = 10, √10000 = 100 (useful for decimals)
- Check your answer: a square root should give a smaller number than you started with
- Use a calculator for complex calculations but know your times tables for mental maths
🧠 Problem-Solving Strategies
Problem-Solving
When estimating roots, always bracket the number between two known square/cube numbers first. For combined operations with powers and roots, apply BIDMAS — indices and roots come before multiplication and addition. Memorise key squares (up to 15²) and cubes (up to 5³) for speed in non-calculator exams.
Multi-Step Problem
A square garden has area 170 m². Estimate the length of each side to 1 decimal place, then use your estimate to approximate the perimeter.
Solution: 13² = 144, 14² = 196. 170 is between 144 and 196, closer to 169 = 13². Since 170 ≈ 169, √170 ≈ 13.0. More precisely, 13² = 144 and 14² = 196, so √170 ≈ 13.0 (actual: 13.04). Perimeter ≈ 4 × 13.0 = 52.0 m.
⚠️ Common Errors
Watch Out!
1. Wrong: √50 = 25 (halving the number) Correct: √49 = 7, √64 = 8, so √50 ≈ 7.1 (between 7 and 8)
2. Wrong: 2³ = 6 (multiplying the base by the power) Correct: 2³ = 2 × 2 × 2 = 8 (multiply the base by itself 3 times)
3. Wrong: √0.64 = 0.32 (halving) Correct: √0.64 = 0.8 (because 0.8 × 0.8 = 0.64)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: (a) Show that 15² − 12² = 81. (b) Use your answer to (a) to find √81. (c) A cube-shaped tank holds 343 litres of water. Find the side length of the tank in cm. (1 litre = 1000 cm³)
(a) 15² = 225, 12² = 144. 225 − 144 = 81 ✓
(b) √81 = 9
(c) Volume = 343 litres = 343,000 cm³. Side = ³√343,000. ³√343 = 7, so ³√343,000 = 70 cm.
Mark scheme: 2 marks for (a) with both squares shown, 1 mark for (b), 3 marks for (c) including unit conversion and correct cube root
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student claims: "If you square a number, the answer is always bigger than the original number."
(a) Give a counter-example to show this claim is false.
(b) For which numbers is the square actually smaller than the original number?
(c) Another student says "If you cube a number greater than 1, the answer is always bigger than the square." Is this correct? Justify your answer.
Answers: (a) 0.5² = 0.25, which is less than 0.5. Or 1² = 1, which equals 1. (b) For numbers between 0 and 1 (not including 0 or 1), the square is smaller. Also, squaring a negative number gives a positive result, which may or may not be "bigger" depending on interpretation. (c) Yes — for n > 1: n³ = n² × n, and since n > 1, n³ > n². For example, 2³ = 8 > 4 = 2².
📝 Exam Questions by Topic
🎬 Video Resources
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