The purpose is not to spend enormous time memorizing every volume. KNOW THE IMPORTANT CUBES SO WELL THAT CUBE RECOGNITION BECOMES IMMEDIATE.
03.0 The Mission
A cube means multiplying a number by itself three times.
n3 = n × n × n
23 = 2 × 2 × 2 = 8
53 = 5 × 5 × 5 = 125
103 = 10 × 10 × 10 = 1000
The Calculation Flow
73 = 7 × 7 × 7
First: 7 × 7 = 49
Then: 49 × 7 = 343
03.1 Why Cubes Matter
Cubes are not as frequently used as squares, but they are valuable across the syllabus. Our goal is 1-20 instant recall.
Simplification
Number System
Powers and indices
Surds and manipulation
Cube roots
Perfect-cube recognition
Mensuration (Volume)
Compound Interest (3 Years)
03.2 The Cube Ladder (1–10)
YOUR FIRST AUTOMATIC CUBE LANDMARKS.
03.3 Cubes 11–20
ABSOLUTE CUBE LANDMARKS. I WANT THESE LIVING IN YOUR MEMORY.
03.4 Cube and Square are Connected
Never learn cubes in isolation. Your previous Square Engine directly supports your Cube Engine. That is why we built the pages in this order.
Calculating 12³
Instead of 12 × 12 × 12, use:
12² × 12
which is:
144 × 12
1440 + 288
1728
Calculating 15³
You already know 15²:
15² × 15
which is:
225 × 15
2250 + 1125
3375
03.5 The Cube Map
THE FULL 1-20 REFERENCE MATRIX
N
N³
N
N³
1
1
11
1331
2
8
12
1728
3
27
13
2197
4
64
14
2744
5
125
15
3375
6
216
16
4096
7
343
17
4913
8
512
18
5832
9
729
19
6859
10
1000
20
8000
03.6 The Last-Digit Pattern
The Secret of Cube Roots
THIS IS EXTREMELY USEFUL FOR RECOGNITION.
Look at the cubes from 1 to 9. Notice something remarkable: The last digit of a cube identifies the last digit of its root.
Ends in 1
Cube 1
→
Root 1
Ends in 2
Cube 8
→
Root 2
Ends in 3
Cube 7
→
Root 3
Ends in 4
Cube 4
→
Root 4
Ends in 5
Cube 5
→
Root 5
Ends in 6
Cube 6
→
Root 6
Ends in 7
Cube 3
→
Root 7
Ends in 8
Cube 2
→
Root 8
Ends in 9
Cube 9
→
Root 9
If a perfect cube is ...512, it ends in 2.
Therefore, its cube root MUST end in 8.
And indeed: 8³ = 512.
03.7 & 03.8 Recognition & The Big Pairs
Squares taught us: 144 → 12. Cubes now teach us: 1728 → 12. Read them both directions.
∛125 = 5
∛216 = 6
∛343 = 7
∛512 = 8
∛729 = 9
∛1000 = 10
∛1331 = 11
∛1728 = 12
∛2197 = 13
∛4096 = 16
∛8000 = 20
03.9 & 03.10 Cube Decomposition
Numbers Ending in 5
For cubes, there isn't an equally simple universal "attach something" shortcut like we had for squares. Don't waste your memory trying to manufacture a fake shortcut. Memorise high-value cubes; calculate unusual cubes through decomposition.
5³ = 125
15³ = 3375
25³ = 15625
35³ = 42875
03.11 Near-Base Cubes
We can extend the same algebraic idea used in our Square Engine when the number is close to a convenient base (like 10).
This will become extremely important later in Number System. A number is a perfect cube when, in its prime factorisation, every prime has an exponent that is a multiple of 3.
216 = 2³ × 3³ (Every exponent is a multiple of 3, therefore 216 = 6³)
1000 = 2³ × 5³ (Therefore 1000 = 10³)
03.15 Cube Root by Last Digit Application
Suppose you are given: ∛12167
1. Look at the last digit: 72. From our table, a cube ending in 7 comes from a root ending in 3.3. Now identify the size (bounding):20³ = 800030³ = 270004. The root is in the 20s and ends in 3.
∛12167 = 23
03.16 Landmarks Above 20
For now, 1–20 is the compulsory memory zone. But you should recognise some larger famous cubes for estimation:
25³ 15625
30³ 27000
40³ 64000
50³ 125000
03.17 The Cube Engine Rule
Squares (1-50) are the major memory target. Cubes (1-20) are the high-value memory bank. Squares occur far more broadly in the calculation ecosystem, so they deserve the larger memory range.
03.18 Self-Test A: Direct Recall
Try these before looking at the answers. Press and hold [?] to reveal.
23
8
43
64
63
216
83
512
103
1000
113
1331
133
2197
153
3375
173
4913
203
8000
03.19 Self-Test B: Reverse Recognition
Find the cube root. Press and hold [?] to reveal.
∛27
3
∛64
4
∛125
5
∛343
7
∛512
8
∛729
9
∛1000
10
∛1728
12
∛2744
14
∛4096
16
03.20 Self-Test C: Last-Digit Logic
Apply the pattern to find the final digit of the cube root.
A perfect cube ends in 8. The cube root ends in:
2
A perfect cube ends in 3. The cube root ends in:
7
A perfect cube ends in 5. The cube root ends in:
5
A perfect cube ends in 9. The cube root ends in:
9
03.21 Master Test
Now mix everything. Enter your answers to test your immediate recall.
123 =
143 =
163 =
183 =
193 =
∛1331 =
∛2197 =
∛3375 =
∛4913 =
∛6859 =
Is 512 a perfect cube?
Is 500 a perfect cube?
What must the last digit of the root be if a perfect cube ends in 6?
What must the last digit of the root be if a perfect cube ends in 2?
Calculate 21³
03.22 The Cube Memory Card
Your mental card should eventually look like this:
1 → 1
2 → 8
3 → 27
4 → 64
5 → 125
6 → 216
7 → 343
8 → 512
9 → 729
10 → 1000
11 → 1331
12 → 1728
13 → 2197
14 → 2744
15 → 3375
16 → 4096
17 → 4913
18 → 5832
19 → 6859
20 → 8000
03.23 Mastery Standard
see 12³ → 1728see 15³ → 3375see 4096 → 16³see 6859 → 19³determine the possible final digit of a cube rootrecognise perfect cubescalculate modest cubes through decompositionuse cube information inside an SSC problem without stopping to rebuild everything