Mathematics 8 · Handsworth 2026–27 · Unit 2
Squares, Cubes & Roots
Ms. Ras Mulinta
Handsworth Secondary
Guided notes
Squaring a number means building a square out of it. Cubing means building a cube. Roots run the idea backwards: given the area of a square, what was the side? This unit is short and very concrete, and it sets up the Pythagorean theorem you will meet later. Each idea has a worked example, then a few problems to try. Answers are at the end of each section so you can check yourself right away.
Name:Block:Date:
The toolkit words we will use
To square a number, multiply it by itself: 5² = 5 × 5 = 25. The small raised 2 is an exponent. A perfect square is what you get when you square a whole number — 1, 4, 9, 16, 25, and so on. To cube a number, use it as a factor three times: 5³ = 5 × 5 × 5 = 125. A perfect cube is what you get when you cube a whole number — 1, 8, 27, 64, and so on.
A square root undoes a square: the square root of 25 is 5, because 5 × 5 = 25. We write it √25 = 5. A cube root undoes a cube: the cube root of 125 is 5, because 5 × 5 × 5 = 125. We write it ∛125 = 5.
Watch the words. "Squared" is times-itself-twice, not times-2. So 5² = 25, not 10. And 5³ = 125, not 15. Read the exponent as how many of that number you multiply together.
1 · Perfect squares a number times itself
A perfect square is the area of a square whose side is a whole number. A 7-by-7 square has area 49, so 49 is a perfect square and 7² = 49. Knowing the first fifteen perfect squares by heart makes the rest of this unit fast.
The first fifteen perfect squares
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64
9² = 81 10² = 100 11² = 121 12² = 144 13² = 169 14² = 196 15² = 225
Worked example — squaring
Find 12².
12² means 12 × 12.
12 × 12 = 144.
= 144
Worked example — is it a perfect square?
Is 90 a perfect square?
9² = 81 and 10² = 100. 90 falls between them, with no whole number landing on it.
No — 90 is not a perfect square.
Check yourself
- Find 6².
- Find 11².
- Is 64 a perfect square? How do you know?
Answers
1. 6 × 6 = 36 2. 11 × 11 = 121 3. Yes — 8 × 8 = 64, so 64 = 8².
2 · Perfect cubes a number used three times
A perfect cube is the volume of a cube whose edge is a whole number. A cube with edge 4 has volume 4 × 4 × 4 = 64, so 64 is a perfect cube and 4³ = 64. Notice 64 is on both lists — it is a perfect square and a perfect cube. Know the first ten cubes and the unit gets easy.
The first ten perfect cubes
1³ = 1 2³ = 8 3³ = 27 4³ = 64 5³ = 125
6³ = 216 7³ = 343 8³ = 512 9³ = 729 10³ = 1000
Worked example — cubing
Find 4³.
4³ means 4 × 4 × 4.
Work in steps: 4 × 4 = 16, then 16 × 4 = 64.
= 64
Worked example — is it a perfect cube?
Is 100 a perfect cube?
4³ = 64 and 5³ = 125. 100 falls between them, with no whole number landing on it.
No — 100 is not a perfect cube.
Check yourself
- Find 3³.
- Find 6³.
- Is 27 a perfect cube? How do you know?
Answers
4. 3 × 3 × 3 = 27 5. 6 × 6 × 6 = 216 (6 × 6 = 36, 36 × 6 = 216) 6. Yes — 3 × 3 × 3 = 27, so 27 = 3³.
3 · Square roots undo the square
A square root answers the question: "what number, times itself, gives this?" Since 8 × 8 = 64, we say √64 = 8. The square-root symbol √ is called a radical. If the number under the radical is a perfect square, the root is a whole number — picture a square with that area and read off its side.
Worked example
Find √144.
Ask: what number times itself is 144?
12 × 12 = 144.
√144 = 12
Worked example — side of a square
A square garden has area 81 m². How long is each side?
Side = √81. Ask: what times itself is 81? 9 × 9 = 81.
Each side is 9 m.
Check yourself
- Find √49.
- Find √100.
- Find √169.
Answers
7. √49 = 7 (7 × 7 = 49) 8. √100 = 10 (10 × 10 = 100) 9. √169 = 13 (13 × 13 = 169)
4 · Cube roots undo the cube
A cube root answers: "what number, used three times, gives this?" Since 5 × 5 × 5 = 125, we say ∛125 = 5. The small 3 on the radical tells you it is a cube root. If the number under the radical is a perfect cube, the root is a whole number — picture a cube with that volume and read off its edge.
Worked example
Find ∛216.
Ask: what number used three times is 216?
6 × 6 × 6 = 216.
∛216 = 6
Worked example — edge of a cube
A cube box has volume 27 cm³. How long is each edge?
Edge = ∛27. Ask: what used three times is 27? 3 × 3 × 3 = 27.
Each edge is 3 cm.
Check yourself
- Find ∛8.
- Find ∛1000.
- Find ∛343.
Answers
10. ∛8 = 2 (2 × 2 × 2 = 8) 11. ∛1000 = 10 (10 × 10 × 10 = 1000) 12. ∛343 = 7 (7 × 7 × 7 = 343)
5 · Estimating roots that are not perfect trap it between two whole numbers
Most numbers are not perfect squares, so their roots are not whole numbers. You can still estimate. Find the two perfect squares it sits between — that traps the root between two whole numbers. Then judge which one it is closer to, and pick a decimal in between.
Worked example — estimate √50
50 sits between 49 and 64. So √50 is between √49 = 7 and √64 = 8.
50 is only 1 past 49, so the root is just above 7 — close to 7.1.
Check: 7.1 × 7.1 = 50.41, very close to 50.
√50 ≈ 7.1
Worked example — estimate √20
20 sits between 16 and 25. So √20 is between √16 = 4 and √25 = 5.
20 is a bit less than halfway from 16 to 25, so the root is a little under 4.5 — about 4.5.
Check: 4.5 × 4.5 = 20.25, very close to 20.
√20 ≈ 4.5
You can estimate cube roots the same way: trap the number between two perfect cubes. For ∛30, since 3³ = 27 and 4³ = 64, the root is just above 3 — about 3.1.
Check yourself
- Between which two whole numbers is √40? Then estimate it.
- Between which two whole numbers is √90? Then estimate it.
- Between which two whole numbers is ∛50?
Answers
13. Between 6 and 7 (since 36 < 40 < 49); 40 is close to 36, so √40 ≈ 6.3 (6.3 × 6.3 = 39.69). 14. Between 9 and 10 (since 81 < 90 < 100); √90 ≈ 9.5 (9.5 × 9.5 = 90.25). 15. Between 3 and 4 (since 27 < 50 < 64).
If you want to go further
Square roots are exactly what you need for the Pythagorean theorem, coming later this year. In a right triangle, a² + b² = c². To find the long side c, you square the two short sides, add, and then take a square root to undo the square. Everything you practised here — squaring, and trapping a root between two whole numbers — is the engine that runs Pythagoras.