⚡ Powers, Roots & Index Notation

Cambridge Lower Secondary Stage 7 · Number & Calculation

Squares & Square Roots
6² = 36  ↔  √36 = 6
5² = 25  ↔  √25 = 5
Index Notation
2⁵ = 2×2×2×2×2 = 32
aⁿ means a multiplied n times
Estimating Roots
7² = 49, 8² = 64
√50 ≈ 7.1  (between 7 and 8)

Watch a perfect square assemble!

What you'll learn:

  • Square numbers and perfect squares up to 15²
  • Cube numbers up to 5³ (and beyond)
  • Index notation: aⁿ meaning and expanded form
  • Square roots and cube roots
  • Powers of 10: 10¹, 10², 10³ …
  • Estimating square roots between consecutive integers
Created by Mr Josh

📖 Learn: Powers, Roots & Index Notation

Part 1: Square Numbers

A square number is the result of multiplying a whole number by itself.

n² = n × n

nn² nn²
11636
24749
39864
416981
52510100

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Part 2: Square Roots

The square root (√) is the inverse of squaring — it asks "what number times itself gives this?"

√36 = 6 because 6 × 6 = 36
√144 = 12 because 12 × 12 = 144

Estimating non-perfect square roots:

√50: we know 7² = 49 and 8² = 64, so √50 is between 7 and 8, closer to 7.  √50 ≈ 7.1
√20: we know 4² = 16 and 5² = 25, so √20 is between 4 and 5.  √20 ≈ 4.5

Part 3: Cube Numbers

A cube number is n × n × n = n³. Imagine filling a cube-shaped box!

nn³Why
111×1×1
282×2×2
3273×3×3
4644×4×4
51255×5×5
10100010×10×10

Cube root: ∛8 = 2, ∛27 = 3, ∛64 = 4, ∛125 = 5

Part 4: Index Notation

Index notation is a short way to write repeated multiplication.

an = a × a × a × … (n times)

Index formExpandedValue
2³2×2×28
3⁴3×3×3×381
5²5×525
10³10×10×101000

Vocabulary: In 2⁵, the base is 2 and the exponent (index) is 5. We say "2 to the power of 5".

Any base to the power 1 = itself: 7¹ = 7
Any base to the power 0 = 1: 5⁰ = 1 (Cambridge rule to know)

Part 5: Powers of 10

10¹ = 10
10² = 100  (1 followed by 2 zeros)
10³ = 1 000  (1 followed by 3 zeros)
10⁴ = 10 000  (1 followed by 4 zeros)
10⁶ = 1 000 000 = one million

Pattern: 10ⁿ is always 1 followed by n zeros!

Created by Mr Josh

💡 Worked Examples

Example 1: Find a Square and its Root

Find 13² and hence write down √169.

Step 1: 13² = 13 × 13
Step 2: 13 × 13 = 169
Step 3: So √169 = 13 (because 13² = 169)

Answer: 13² = 169, √169 = 13

Example 2: Write in Index Notation

Write 5 × 5 × 5 × 5 in index notation and find its value.

Step 1: Count how many 5s: there are 4 fives.
Step 2: Index notation: 5⁴
Step 3: Value: 5 × 5 = 25, 25 × 5 = 125, 125 × 5 = 625

Answer: 5⁴ = 625

Example 3: Estimate √60

Between which two consecutive integers does √60 lie? Estimate to 1 d.p.

Step 1: Find the nearest perfect squares: 7² = 49, 8² = 64
Step 2: 49 < 60 < 64, so 7 < √60 < 8
Step 3: 60 is closer to 64 than 49 (gap of 4 vs gap of 11), so √60 ≈ 7.7

Answer: √60 is between 7 and 8; √60 ≈ 7.7

Example 4: Cube Root Problem

A cube has volume 125 cm³. Find the side length.

Step 1: Volume of a cube = side³, so side³ = 125
Step 2: Find ∛125: ask "what number × itself × itself = 125?"
Step 3: 5 × 5 × 5 = 125 ✓

Answer: side length = 5 cm

Created by Mr Josh

🔭 Interactive Visualizer

Square Grid Builder

Click a number to watch the n×n grid assemble, then click "Take Root" to collapse it!

Cube Builder (n³)

Drag the slider to change n and see n³!

n = 1, n³ = 1

Square Root Estimator

Use the slider to estimate √50. Then reveal the exact value!

Your estimate: —

Index Notation Builder

Use + / − to set the base and exponent, then watch the expanded form!

Base
2
Exponent
3
2³
2 × 2 × 2 = 8

Powers Memory Match

Flip cards to match each power in index form with its value!

Created by Mr Josh

✏️ Exercise 1 — Squares & Square Roots

Calculate each square and square root. Write your answer in the box.

Created by Mr Josh

✏️ Exercise 2 — Cubes & Cube Roots

Calculate each cube and cube root.

Created by Mr Josh

✏️ Exercise 3 — Index Notation

Write in index form or evaluate. Use the format: base^exponent (e.g. 2^4 for 2⁴).

Created by Mr Josh

✏️ Exercise 4 — Estimating Square Roots

For each question, write the two consecutive integers the root lies between, then estimate to 1 d.p.

  1. √10: between 3 and 4; √10 ≈ 3.2
  2. √27: between 5 and 6; √27 ≈ 5.2
  3. √45: between 6 and 7; √45 ≈ 6.7
  4. √72: between 8 and 9; √72 ≈ 8.5
  5. √55: between 7 and 8; √55 ≈ 7.4
  6. √130: between 11 and 12; √130 ≈ 11.4
Created by Mr Josh

✏️ Exercise 5 — Powers of 10 & Mixed

Evaluate each expression.

Created by Mr Josh

📝 Practice — 20 Mixed Questions

  1. Find 9²
  2. Find 4³
  3. Find √64
  4. Find ∛27
  5. Write 3 × 3 × 3 × 3 × 3 in index notation.
  6. Evaluate 2⁶
  7. Write 10⁵ as an ordinary number.
  8. Between which two consecutive integers does √40 lie?
  9. Estimate √80 to 1 decimal place.
  10. Find 12²
  11. Find ∛125
  12. Write 7⁴ in expanded form and evaluate.
  13. Which is bigger: 3⁴ or 4³? Show working.
  14. Find √225
  15. What is 2⁰?
  16. Find 11²
  17. Between which two integers does √110 lie?
  18. Evaluate 5³ − 10²
  19. Write 100 000 as a power of 10.
  20. A square has area 196 cm². Find its side length.
  1. 81
  2. 64
  3. 8
  4. 3
  5. 3⁵
  6. 64
  7. 100 000
  8. 6 and 7
  9. 8.9
  10. 144
  11. 5
  12. 7×7×7×7 = 2401
  13. 3⁴ = 81; 4³ = 64; so 3⁴ is bigger.
  14. 15
  15. 1
  16. 121
  17. 10 and 11
  18. 125 − 100 = 25
  19. 10⁵
  20. Side = √196 = 14 cm
Created by Mr Josh

🏆 Challenge Questions

  1. A square courtyard has area 289 m². A square fountain in its centre has side 4 m. Find the area of courtyard NOT covered by the fountain.
  2. Two cubes: Cube A has side 3 cm, Cube B has side 4 cm. How much greater is the volume of Cube B than Cube A?
  3. Show that 2¹⁰ = 1024 by writing out the powers step by step: 2¹, 2², 2³, …, 2¹⁰.
  4. Leila says "√81 + √16 = √97". Is she correct? Explain why or why not.
  5. Find the smallest integer n such that n² > 200.
  6. A cube has volume 512 cm³. A square tile has side equal to the cube's side length. Find the area of the tile.
  7. Write 8 as a power of 2 and as a power with exponent 3. Hence write 8³ as a single power of 2.
  8. Explain why every even perfect square is divisible by 4. Use examples n = 2, 4, 6 to support your answer.
  1. 289 − 16 = 273 m²
  2. 4³ = 64, 3³ = 27; difference = 37 cm³
  3. 2,4,8,16,32,64,128,256,512,1024 → 2¹⁰ = 1024 ✓
  4. No. √81 = 9, √16 = 4, 9+4 = 13. But √97 ≈ 9.85. Addition under different roots ≠ sum of roots.
  5. 14² = 196 ≤ 200; 15² = 225 > 200. Answer: n = 15
  6. ∛512 = 8; tile area = 8² = 64 cm²
  7. 8 = 2³; 8³ = (2³)³ = 2⁹ = 512
  8. (2k)² = 4k² for any integer k, so always divisible by 4. e.g. 2²=4=4×1; 4²=16=4×4; 6²=36=4×9 ✓
Created by Mr Josh