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Cube Numbers

Introduction

Consider that there are eight blue and seven yellow cubes in each of the two sets of cubes.

Try arranging these cubes by creating bigger cubes. The illustration below demonstrates how the eight little blue cubes combine to form a bigger cube. No matter how many times we arrange the seven yellow cubes, we are unable to create a larger cube.

This example demonstrates that 8 is a cube number, whereas 7 is not. Using the same number of smaller cubes, a cube number is any number that may be formed into a cube. Aside from 8, there are an infinite number of cube numbers. There are 27, 64, 125, 216, 343, 512, 729, 1000 and so forth.

We will study more about cube numbers, their characteristics, and a few examples with solutions in this article.

What are cube numbers?

Definition

A cube number is an outcome of multiplying a number by itself and then multiplying it by itself again. In essence, multiplying three same numbers yields a cube number. A perfect cube is a numerical value that may be obtained by cubing any given number. 

Some examples are 1, 8, 27, 64, 125, 216, 343 and so forth.

1 cubed is 1 x 1 x 1 =1
2 cubed is 2 x 2 x 2 =8
3 cubed is 3 x 3 x 3=27
4 cubed is 4 x 4 x 4 =64
5 cubed is 5 x 5 x 5 =125
6 cubed is 6 x 6 x 6=216
7 cubed is 7 x 7 x 7=343
8 cubed is 8 x 8 x 8=512
9 cubed is 9 x 9 x 9=729
10 cubed is 10 x 10 x 10=1000

Application

Using cube blocks, a cube number is any number that may be formed into a cube using smaller cubes. These are some examples of cube numbers.

The red cube has a total of 8 smaller cubes, while the yellow cube has a total of 27 smaller cubes. Notice that the red cube has each side that measures 2 units while the yellow cube has 3 units on each side. This also explains a practical application of cubic numbers, that is, getting the volume of 3D shapes.

Hence, the volume of each cube can be calculated by multiplying the lengths of their sides to themselves three times. In geometry, a cube’s volume equals the product of its length,  width,  and height.

In the solution below, notice the use of the exponent3” or power of 3. Since 2 for the red cube’s volume is multiplied to itself three times the same as the number 3 for the volume of the yellow cube. Saying 23, 2 cubed, is the same as 2×2×2. Saying 33, 3 cubed is the same as 3×3×3. 

Volume Red Cube =2×2×2=23
Volume ( Red Cube=8 cubic units

Volume Yellow Cube =3×3×3=33
Volume Yellow Cube =27 cubic units

In metric measures, cubic meters, cubic centimetres, and cubic inches, cubic feet are a few examples of cubic units. Thus, if, for example, the red cube has each side measures 2 centimetres, and the yellow cube has each side that measures 3 centimetres, we have,

Volume Red Cube =8 cm3 or 8 cubic centimetres

Volume Yellow Cube =27 cm3 or 27 cubic centimetres

Once you have fully understood cube numbers, working on this backwards gives us the concept of cube roots and offers practical, real-life application.

For example, if you wish to find the volume of a cubic container, then cubing its side length is the solution. If, for instance, you know the volume of the cubic container and you want to find the area of its surface, it is helpful to get the side lengths by getting the cube root of the volume. Since you already know the dimension of the cubic container, you can now quickly compute the amount of paint needed. 

List of Cubes 1 to 100 

The cubes 1 to 100 or the cubes of the first 100 numbers are displayed in the tables below and can be used to solve related word problems.

Cubes 1 to 25
13=163=216113=1331163=4096213=9261
23=873=343123=1728173=4913223=10648
33=2783=512133=2197183=5832233=12167
43=6493=729143=2744193=6859243=13824
53=125103=1000153=3375203=8000253=15625
Cubes 26 to 50
263=17576313=29791363=46656413=68921463=97336
273=19683323=32768373=50653423=74088473=103823
283=21952333=35937383=54872433=79507483=110592
293=24389343 =39304393=59319443=85184493=117649
303=27000353=42875403=64000453=91125503=125000
Cubes 51 to 75
513=132651563=175616613=226981663=287496713=357911
523=140608573=185193623=238328673=300763723=373248
533=148877583=195112633=250047683=314432733=389017
543=157464593=205379643=262144693=328509743=405224
553=166375603=216000653=274625703=343000753=421875
Cubes 76 to 100
763=438976813=531441863=636056913=753571963=884736
773=456533823=551368873=658503923=778688973=912673
783=474552833=571787883=681472933=804357983=941192
793=493039843=592704893=704969943=830584993=970299
803=512000853=614125903=729000953=8573751003=1000000

Interesting Cube Number Facts

A number that is even will produce an even number when it is cubed.

For example, the cube of 6 is 216, the cube of 12 is 1728, and the cube of 14 is 2744. 

The numbers 6, 12 and 14 are all even numbers, and their cubes are also even numbers.

A number that is odd will produce an even number when it is cubed.

For example, the cube of 4 is 64, the cube of 11 is 1331, and the cube of 45 is 91125.

The numbers 3, 11 and 45 are all odd numbers, and their cubes (27, 1331, and 91125, respectively) are all even numbers.

A positive number’s cube is positive. 

For example, 8×8×8=512; both 8 and 512 are even numbers. 

Another example, 12×12×12=1728; both 12 and 1728 are even numbers.

A negative number’s cube is a negative number.

For example, ( -3 ) x ( -3 ) x ( -3 ). Given that this involves multiplying negative numbers, it is important to keep in mind that a positive number is produced when two positive numbers are multiplied, whereas the product of two numbers with a different sign is negative. Thus,

-33= ( -3 ) x ( -3 ) x ( -3 ) =9 × ( -3 ) = -27

Another example is the cube of -14.

-143=( -14 ) x ( -14 ) x ( -14 )=196 × ( -14 ) = -2744

When a cube number contains zeroes at the end, the total number of zeroes is a multiple of 3.

When we say the total of zeroes is a multiple of 3, we are referring to 3, 6, 9, 12, and so on.

Some examples are 8000, 64000, and 1000000. 

20×20×20=8000  (With 3 zeroes)

40×40×40=64000 (With 3 zeroes)

100×100×100=1000000 (With 6 zeroes)

If you cube numbers with the last digits 1, 4, 5, 6, or 9, the result will also end with the same digit.

Examples are,

The cube of the numbers 11, 21, 31, and 41 are 1331, 9261, 29791, and 68921, respectively. 11, 21, 31, and 41 and their cubes end in 1 too.

The cube of the numbers 16, 26, 36, and 46 are 4096, 17576, 46656, and 97336, respectively. 16, 26, 36, and 46 and their cubes end in 6 too.

Patterns of Cube Numbers

A sum of consecutive odd numbers is a perfect cube. The series of odd numbers that, when added together, form a perfect cube are found in the number line’s odd number order.

In reference to the below, the first column is the consecutive odd numbers, the second column is the sum of a consecutive odd number, and the third column is the equivalent cube numbers.

1113
3 + 5823
7 + 9 + 112733
13 + 15 + 17 + 196443
21 + 23 + 25 + 27 + 2912553
31 + 33 + 35 + 37 +39 + 41 21663
43 + 45 + 47 + 49 + 51 + 53 + 5534373

The number of consecutive odd numbers whose sum forms a perfect cube is equal to the number being considered for the cube.

For example, for 4 cubed, adding the 4 odd numbers, 13, 15, 17 and 29 gives us 64. Same with 5 cubed, adding the 5 consecutive numbers 21 + 23 + 25 + 27 + 29 will give us 125.

11 odd number13
3 + 52 odd numbers23
7 + 9 + 113 odd numbers33
13 + 15 + 17 + 194 odd numbers43
21 + 23 + 25 + 27 + 295 odd numbers53
31 + 33 + 35 + 37 +39 + 41 6 odd numbers63
43 + 45 + 47 + 49 + 51 + 53 + 557 odd numbers73

Examples

Example 1

Find the cube number of the number 43.

Solution:

To find the cube number of the number 43, we must multiply 43 by itself and then multiply it to itself again, so we have 433.

433=43 x 43 x 43 = 433= 79507

To check the answer, you may use the list of cube numbers provided earlier. Thus, the cube of 43 is 79507.

Example 2

Find the cube number of 101.

Solution:

To find the cube number of the number 101, we must multiply 101 to itself and then multiply it to itself again, so we have 1013.

1013=101×101×101 = 1013= 1030301

Thus, the cube of 101 is 1030301.

Example 3

Identify the cube numbers.

a.  125 

b.  1331

c.  2750

d. 5832

e.  8000

Solution:  Only 2750 is not a cube number among the given numbers. 

125 is a cube number of 5 or 53 = 125. 

1331 is a cube number of 11 or 113 = 1331.

5832 is a cube number of 18 or 183 = 5832.

And 8000 is a cube number of 20 or 203 = 8000.

Example 4

Write each given using the power of 3 or the cubed symbol, then evaluate.

a. 9×9×9

b. 11×11×11

c. (-4)×(-4)×(-4)

d. 12×12×12

e. 20×20×20

f. -25 -25 ×( -25 )

Solution:

a. 9×9×9=93=729

b. 11×11×11=113=1331

c. ( -4 ) x ( -4 ) x ( -4 ) = ( -4 )3=-64

d. 12×12×12=123=1728

e. 20×20×20=203=8000

f. ( -25 ) x ( -25 ) x ( -25 ) = -15625

Example 5

Solve: 633 +213

Solution:  We must get the cube of 63 and 21 as the first step. The cube number of 63 is 633, which is equal to 250047. The cube number of 21 is 213, which is equal to 9261. 

633+213= ( 63×63 ×63 ) + ( 21×21×21 ) = 250047 + 9261 = 259308

Thus, 633 +213 is equal to 259308.

Example 6

Look for two cube numbers whose sum is 9603.

Solution:

In reference to the list of cube numbers presented earlier, the two cube numbers that give us a sum of 9603 are 2744 and 6859.

2744 is a cube number of 14, while 6859 is a cube number of 19. 

9603=2744+6859=143+193

Example 7

Up to 1000, what are the cube numbers?

Solution:

1 cubed is 1×1×1=13=1

2 cubed is 2×2×2=23=8

3 cubed is 3×3×3=33=27

4 cubed is 4×4×4=43=64

5 cubed is 5×5×5=53=125

6 cubed is 6×6×6=63=216

7 cubed is 7×7×7=73=343

8 cubed is 8×8×8=83=512

9 cubed is 9×9×9=93=729

10 cubed is 10×10×10=103=1000

Thus, the cube numbers up to 1000 are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Example 8

Victoria has wooden cubes, as shown below. If the side length of each cube measures 2 inches, find the total volume of the cubes.

Solution:

First, we need to solve for the volume of each cube, that is,

Volume of each cube=2 inches×2 inches×2 inches
Volume of each cube=8 cubic inches or 8 in3

To get the total volume of the cubes, we simply need to add the volume of each cube, or we multiply 8 in3 by 8.

So, we have, 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64  or

8 in3 × 8 = 64 in3

Hence, the total volume of the cubes is 64 cubic inches.

We may also try forming the 8 small cubes to form a bigger cube. The illustration below shows that the side length of the bigger cube is 4 inches. 

Therefore, to get the volume of this cube with a side length of 4 inches, we have,

Volume=4×4×4
Volume=16×4
Volume=64 in3

Example 9

From 1 to 1000, find cube numbers that are also square numbers.

Solution:

The table below shows the cube numbers from 1 to 1000.

13=163=216
23=873=343
33=2783=512
43=6493=729
53=125103=1000

Now, we need to choose among these cube numbers are square numbers too. So, we have,

12=182=64272=729

Hence, from 1 to 1000, the cube numbers that are also square numbers are 1, 64, and 729.

Summary

A cube number is an outcome of multiplying a number by itself and then multiplying it by itself again. In essence, multiplying three same numbers yields a cube number.

The shorthand way of writing a cube number is to use the exponent “3” or the power of 3. 

The volume of each cube can be calculated by multiplying the lengths of their sides to themselves three times. In geometry, a cube’s volume equals the product of its length, width, and height.

A number that is even will produce an even number when it is cubed.

A number that is odd will produce an even number when it is cubed.

A positive number’s cube is positive. 

A negative number’s cube is a negative number.

When a cube number contains zeroes at the end, the total number of zeroes is a multiple of 3.

If you cube numbers with the last digits 1, 4, 5, 6, or 9, the result will also end with the same digit.

A sum of consecutive odd numbers is a perfect cube. The series of odd numbers that, when added together, form a perfect cube are found in the number line’s odd number order.

The number of consecutive odd numbers whose sum forms a perfect cube is equal to the number being considered for the cube.

Frequently Asked Questions (FAQs)

How to cube a number?

To cube a number, you must multiply it by itself, then by itself again. 

So, for example, to cube the number 6, we have 6×6×6 or 63, which results in 216. 

63=6×6×6=36×6=216

Another example is the 783, so we have to multiply 78 by 78 and their product by 78. Thus,

783=78×78×78=6084×78=474552

What is an application of cube numbers?

When determining the volume of cubes, cube numbers are especially helpful. Since a cube is a three-dimensional shape with sides that are all the same size (length, width, and height), it’s easy to calculate the volume by “cubing” one of its sides.

For example, if we have a cube with a side length equal to 4 units, we will have a volume of 64 cubic units. This is done by cubing the number 4. Here is the solution,

43=4×4×4=16×4=64 cubic units

Another example is if we have specific metric units like a cube with a side length of 9 centimetres. If we cube the number 9, we will be getting its volume. That is,

93=9×9×  9=81×9=729 cm3

What differentiates square numbers from cube numbers?

A number is multiplied by itself once to get a square number, while the operation is done twice to form a cube number.

For instance, let us use the number 6. 36 is the result of multiplying 6 by itself. Hence, 36 is the square of 5. 

6×6=36  or

62=36

Notice the use of the exponent “2” getting the square of the number.

From here, multiplying 62 by 6 again gives us the cube of 6 or 63. Therefore,

63=6×6×6=36×6=216

How to find the cube of a negative number?

A negative number’s cube is a negative number.

For example, ( -3 ) x ( -3 ) x ( – 3 ). Given that this involves multiplying negative numbers, it is important to keep in mind that the product of two numbers with the same sign is positive, whereas the product of two numbers with a different sign is negative. Thus,

( -3 )3=( -3 ) x ( -3 ) x ( – 3 ) = 9× ( -3 ) = -27

Another example is the cube of -14.

-143=( -14 ) x ( -14 ) x ( -14 )=196 × ( -14 )=-2744

What are the first 10 cube numbers?

These are the first ten cube numbers: 1, 8, 27, 64, 125, 216, 343-512, 729, and 1000.

Cubes1 to 10
13=163=216
23=873=343
33=2783=512
43=6493=729
53=125103=1000

What is a cube number written in shorthand?

The shorthand way of writing a cube number is to use the exponent “3” or the power of 3. 

Examples below show the cubes of 1 to 10 in their shorthand and expanded form writing.

1 cubed is 1×1×1=13=1

2 cubed is 2×2×2=23=8

3 cubed is 3×3×3=33=27

4 cubed is 4×4×4=43=64

5 cubed is 5×5×5=53=125

6 cubed is 6×6×6=63=216

7 cubed is 7×7×7=73=343

8 cubed is 8×8×8=83=512

9 cubed is 9×9×9=93=729

10 cubed is 10×10×10=103=1000

How many are even cube numbers there from 1 to 1000?

There are 5 even cube numbers from 1 to 1000. These numbers are 8, 64, 216, 512, and 1000.

How many are odd cube numbers there from 1 to 1000?

There are 5 odd cube numbers from 1 to 1000. These numbers are 1, 27, 125, 343, and 729.

How do you identify if a number is a perfect cube or not?

A number can be determined to be a perfect cube or not using the prime factorization method. Let us say, for example, let us identify if 512 and 729 and 1728 are perfect cubes using prime factorization.

We will obtain 2 × 2 × 2  × 2 × 2 × 2 × 2 × 2 × 2 as the prime factors of 512.

We can make 3 groups of 2 cubes, and we have (2×2×2)×(2×2×2)×(2×2×2) or 23 x 23 x 23.

Thus, 512=( 2×2×2 )3=83.

For 729, the prime factorization is 3×3×3×3×3×3.

We can make 2 groups of 3 cubes from the prime factors, hence,

 ( 3×3×3 )×( 3×3×3 ) or  33 x 33

Therefore, 729= ( 3×3 )3=93.

Now, to show if 1728 is a perfect cube, let us also have its prime factorization.

The prime factorization of 1728 is 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3.

We can make 3 groups with each group having 2×2×3 as factors, so we have,

 (2 × 2 × 3)3 ×(2 × 2 × 3) ×(2 × 2 × 3).

Hence, 1728= (2×2×3)3=123.

Therefore, the numbers 512, 729 and 1928 are all perfect cubes.

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