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
🧮 Worked Examples — Exam Style
Square and cube numbers are free marks — but only if you have memorised them up to 15² and 5³. Estimation questions reward bracketing between perfect squares.
A square garden has an area of 169 m². Find the length of one side.
Side × side = 169, so side = √169.
Check nearby squares: 12² = 144 (too small), 13² = 169 ✓
Answer: 13 m
B1 for 13. Units matter: m not m² — the side is a length.
Estimate √90
Bracket between perfect squares: 81 < 90 < 100, so 9 < √90 < 10.
90 is closer to 81 than to 100? 90 − 81 = 9 and 100 − 90 = 10, so try ≈ 9.5.
9.5² = 90.25 — just over, so √90 ≈ 9.49 (accept 9.4–9.5).
M1 for 81 < 90 < 100, A1 for an estimate between 9.4 and 9.5. The answer to 1 d.p. is 9.5.
Exam wording to expect: "Write down the value of …" (memorised squares/cubes), or "Estimate …" (never a calculator answer).
❓ 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
Q6: Calculate: √225
Q7: Write down the first five square numbers
✅ 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
🧮 Worked Examples — Harder Exam Style
A cube has a volume of 216 cm³. Find the length of one edge.
Edge × edge × edge = 216, so edge = ³√216
Try 5: 5³ = 125 (too small). Try 6: 6 × 6 × 6 = 216 ✓
Answer: 6 cm
B1 for 6. Cube volumes reversing to edge length is the standard cube-root context at Foundation.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 7² = 7 × 7 = 49; 11² = 11 × 11 = 121
Q2. √81 = 9 (9 × 9 = 81); √196 = 14 (14 × 14 = 196)
Q3. 6³ = 6 × 6 × 6 = 216; ³√125 = 5 (5 × 5 × 5 = 125)
Q4. 7² = 49 and 8² = 64, and 50 sits almost exactly between them: √50 ≈ 7.1 (actual 7.07…). Method mark for using 49 < 50 < 64.
Q5. 2⁵ = 32 and 3⁴ = 81, so 32 − 81 = −49. Do each power FIRST, then subtract — 2⁵⁻⁴ is the classic misread.
Q6. √225 = 15
Q7. 1, 4, 9, 16, 25
📌 Key Facts to Memorise
- Squares up to 15²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cubes up to 5³: 1, 8, 27, 64, 125 (and 10³ = 1000)
- Square root undoes a square: √(a²) = a
- Estimating roots: bracket between two perfect squares, then narrow down
- Powers of 10: 10⁴ = 10,000 (number of zeros = index)
🗺️ In the Exam
Squares/cubes recall earns 1 mark on Paper 1 nearly every year. Estimation of roots is a 2-mark non-calculator favourite. Powers feed standard form and index laws.
Links to: index laws (N7), standard form (N9), Pythagoras (G), area problems (G).
⚡ Targeted Drills
Thirty-second checks — cover the answers, work each one, then mark yourself. If any takes longer than a minute, revisit the matching section above.
- 12² → 144 · 13² → 169 · 4³ → 64
- √144 → 12 · ³√64 → 4 · √400 → 20
- Between which two square numbers does √70 sit? → 64 (8²) and 81 (9²), so ≈ 8.4
- 10⁵ → 100,000 · 2⁶ → 64
- A square has area 196 cm². Perimeter? → side √196 = 14, perimeter 4 × 14 = 56 cm
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 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.
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🎬 Video Resources
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