What are Cube Numbers?

What are Cube Numbers?

videolesson.JPG

The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

list of the first 10 cube numbers

A cube number is the result of multiplying a number by itself twice. For example 23 = 2 × 2 × 2 = 8.

definition of a cube number

  • When any whole number is multiplied by itself twice a cube number is formed.
  • For example, 2 cubed means 2 × 2 × 2 which equals 8.
  • Therefore we say that 8 is a cube number.
  • ‘Cubing’ a number means to multiply a number by itself twice.
  • We write a small 3 above the number to tell us to cube it like so: 23.
  • Cube numbers are called this because cubing a side length of a cube gives us the volume of a cube.
  • If a cube has a side length that is a whole number, its volume will be a cube number.

videolesson.JPG

example of calculating cube numbers

  • To cube a number, multiply it by itself twice.
  • 2 cubed is written as 23.
  • 2 × 2 × 2 = 8 and so, 2 cubed is 8.
  • 3 cubed is written as 33.
  • 3 × 3 × 3 = 27 and so, 3 cubed is 27.
  • 4 cubed is written as 43.
  • 4 × 4 × 4 = 64 and so, 4 cubed is 64.

practiseqs.JPG

Cube Numbers Activity

Cube Numbers Activity

Cube Numbers Worksheets and Answers

Cube Numbers

What are Cube Numbers?

A cube number is formed when any whole number is multiplied by itself twice. The symbol for cubing a number is 3. For example, 2 cubed or 23 means 2 × 2 × 2 which equals 8. Therefore 8 is a cube number. Cube numbers are called this because the volume of a cube is found by cubing its side length.

Below shows a cube with side lengths of 2 cm.

The volume of the cube is found by multiplying a side length by itself twice.

a cube number is the volume of a cube

2 × 2 × 2 = 8 and so the volume of the cube is 8 cm3.

2 cubed is 8 shown visually

This means that the overall cube is made up of 8 smaller 1 cm3 cubes.

8 is a cube number because it is formed by multiplying a whole number by itself twice.

2 × 2 × 2 = 8

We can write this more simply as 23 = 8.

The number is written to the power of 3, which is a shorter way of writing the full multiplication.

How to Find a Cube Number

To find a cube number, multiply any whole number by itself and then by itself again. The easiest way to do this is to do each multiplication separately. For example, 3 cubed is 3 × 3 × 3. The first multiplication is 3 × 3 = 9 and then the second multiplication is 9 × 3 = 27. Therefore the 3rd cube number is 27.

3 cubed is 27

Here is another example of calculating 4 cubed.

43 means 4 × 4 × 4.

The first step is to multiply 4 × 4 to get 16.

The next step is to multiply 16 by 4 again.

16 × 4 = 64.

4 cubed is 64

64 is the 4th cube number.

List of Cube Numbers

The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.

cube numbers list

Here is a complete list of the first 100 cube numbers:

Number Calculation Cube Number
131 × 1 × 1 =1
232 × 2 × 2 =8
333 × 3 × 3 =27
434 × 4 × 4 =64
535 × 5 × 5 =125
636 × 6 × 6 =216
737 × 7 × 7 =343
838 × 8 × 8 =512
939 × 9 × 9 =729
10310 × 10 × 10 =1000
11311 × 11 × 11 =1331
12312 × 12 × 12 =1728
13313 × 13 × 13 =2197
14314 × 14 × 14 =2744
15315 × 15 × 15 =3375
16316 × 16 × 16 =4096
17317 × 17 × 17 =4913
18318 × 18 × 18 =5832
19319 × 19 × 19 =6859
20320 × 20 × 20 =8000
21321 × 21 × 21 =9261
22322 × 22 × 22 =10648
23323 × 23 × 23 =12167
24324 × 24 × 24 =13824
25325 × 25 × 25 =15625
26326 × 26 × 26 =17576
27327 × 27 × 27 =19683
28328 × 28 × 28 =21952
29329 × 29 × 29 =24389
30330 × 30 × 30 =27000
31331 × 31 × 31 =29791
32332 × 32 × 32 =32768
33333 × 33 × 33 =35937
34334 × 34 × 34 =39304
35335 × 35 × 35 =42875
36336 × 36 × 36 =46656
37337 × 37 × 37 =50653
38338 × 38 × 38 =54872
39339 × 39 × 39 =59319
40340 × 40 × 40 =64000
41341 × 41 × 41 =68921
42342 × 42 × 42 =74088
43343 × 43 × 43 =79507
44344 × 44 × 44 =85184
45345 × 45 × 45 =91125
46346 × 46 × 46 =97336
47347 × 47 × 47 =103823
48348 × 48 × 48 =110592
49349 × 49 × 49 =117649
50350 × 50 × 50 =125000
51351 × 51 × 51 =132651
52352 × 52 × 52 =140608
53353 × 53 × 53 =148877
54354 × 54 × 54 =157464
55355 × 55 × 55 =166375
56356 × 56 × 56 =175616
57357 × 57 × 57 =185193
58358 × 58 × 58 =195112
59359 × 59 × 59 =205379
60360 × 60 × 60 =216000
61361 × 61 × 61 =226981
62362 × 62 × 62 =238328
63363 × 63 × 63 =250047
64364 × 64 × 64 =262144
65365 × 65 × 65 =274625
66366 × 66 × 66 =287496
67367 × 67 × 67 =300763
68368 × 68 × 68 =314432
69369 × 69 × 69 =328509
70370 × 70 × 70 =343000
71371 × 71 × 71 =357911
72372 × 72 × 72 =373248
73373 × 73 × 73 =389017
74374 × 74 × 74 =405224
75375 × 75 × 75 =421875
76376 × 76 × 76 =438976
77377 × 77 × 77 =456533
78378 × 78 × 78 =474552
79379 × 79 × 79 =493039
80380 × 80 × 80 =512000
81381 × 81 × 81 =531441
82382 × 82 × 82 =551368
83383 × 83 × 83 =571787
84384 × 84 × 84 =592704
85385 × 85 × 85 =614125
86386 × 86 × 86 =636056
87387 × 87 × 87 =658503
88388 × 88 × 88 =681472
89389 × 89 × 89 =704969
90390 × 90 × 90 =729000
91391 × 91 × 91 =753571
92392 × 92 × 92 = 778688
93393 × 93 × 93 =804357
94394 × 94 × 94 =830584
95395 × 95 × 95 =857375
96396 × 96 × 96 =884736
97397 × 97 × 97 =912673
98398 × 98 × 98 =941192
99399 × 99 × 99 =970299
1003100 × 100 × 100 =1000000

Properties of Cube Numbers

Cube numbers have the following properties:

  • An even number cubed is always even.
  • An odd number cubed is always odd.
  • A positive number cubed is always positive.
  • A negative number cubed is always negative.
  • Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit.
  • The sum of the cubes of the first n natural numbers is equal to the square of their sum.

Here are some examples to illustrate these properties of cube numbers.

An even number cubed is always even

For example, 2 is an even number. If we cube it we get 8, which is an even answer.

2 × 2 × 2 = 8

This property works because if any number is multiplied by an even number at least once, the result is even. Cubing an even number means that we must multiply by an even number.

An odd number cubed is always odd

For example, 3 is an odd number. If we cube it we get 27, which is an odd answer.

3 × 3 × 3 = 27

This property works because if two odd numbers are multiplied together, the result is always odd. To make an even number, at least one of the numbers being multiplied will need an even factor. However, when an odd number is cubed, we have odd × odd × odd and so, no factor of two appears in the final result.

A positive number cubed is always positive

For example, 10 is a positive number. If we cube it we get 1000, which is also positive.

10 × 10 × 10 = 1000

This property works because a negative number can only be made from another negative. If we only multiply positive numbers, the result must be positive.

A negative number cubed is always a negative

For example, (-2) × (-2) × (-2) = -8.

When we cube a number, we multiply it by itself twice. If a negative number is cubed, we have three negative numbers multiplied together.

When three negative numbers are multiplied together, the result is always negative.

(-2) × (-2) = +4 and then 4 × (-2) = -8.

Cubing a number ending in 0, 1, 4, 5, 6 or 9 will result in a number ending in this same digit

For example:

103 = 1000. Both numbers end in 0.

213 = 9261. Both numbers end in 1.

43 = 64. Both Numbers end in 4.

153 = 3375. Both numbers end in 5.

663 = 287496. Both numbers end in 6.

193 = 6859. Both numbers end in 9.

The sum of the cubes of the first n natural numbers is equal to the square of their sum

13 + 23 + 33 + … + n3 = (1 + 2 + 3 + … + n)2.

For example, for an n of 5:

13 + 23 + 33 + 43 + 53 = 225.

(1 + 2 + 3 + 4 + 5)2 = 225.

Cubing the consecutive numbers and then adding them up gives the same result as adding the numbers up and then squaring them.

For example, for an n of 10:

13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 = 3025.

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)2 = 3025.

Cube Numbers That Are Also Square Numbers

The cube numbers that are also square numbers are found by raising whole numbers to the power of 6. For example, 16 = 1, 26 = 64 and 36 = 729. 1 is 12 and 13, 64 is 82 and 43 and 729 is 272 and 93.

13 = 1 = 12 23 = 8 33 = 27 43 = 64 = 82 53 = 125 63 = 216 73 = 343 83 = 512 93 = 729 = 272 103 = 1000

1, 64 and 729 are cube numbers and also square numbers.

Further cube numbers that are also square numbers are found by raising any integer to the power of 6:

16 = 1 26 = 64 36 = 729 46 = 4096 56 = 15625 66 = 46656 76 = 117649 86 = 262144 96 = 531441 106 = 1000000

nextlesson.JPG

Now try our lesson on Finding Prime Numbers to 100 where we learn how to find prime numbers.