What is a cube number: What are cube numbers?
Posted onWhat are cube numbers?
Here you can find out what cube numbers are, why they are called cube numbers and how you can help children to understand cube numbers as part of their maths learning at home.
What is a cube number?
A cube number is found when we multiply a number by itself and then itself again. The symbol for cubed is ^{3}.
For example, 8 is a cube number because it’s 2 x 2 x 2; this is also written as 2^{3} (“two cubed”).
Another example of a cube number is 27 because it’s 3^{3} (3 x 3 x 3, or “three cubed”).
A cube number can also be called a number cubed.
 2³ = 2 × 2 × 2 = 8
 3³ = 3 × 3 × 3 = 27
 4³ = 4 × 4 × 4 = 64
 5³ = 5 × 5 × 5 = 125
Cube numbers
Watch this video to find out how to teach cube numbers and square numbers to primary aged children.
As you can see when you cube a whole number, you’ll find the numbers get very big very quickly!
Why are they called cube numbers
They are named cube numbers (or cubed numbers) because they can also be used to calculate the volume of a cube: since a cube is a 3d shape with sides of the same length, width and height, you calculate its volume by multiplying the side length by itself and then itself again (or ‘cubing’ it).
As cubes have equal sides (length, height and width), calculating the volume is simple – just “cube” one of its sides!
For example, a cube with side length 2cm would have a volume of 8cm3 (as 23 = 8). In reverse, if we knew a cube had a volume of 27cm3, we’d know that each side would measure 3cm (as 33 = 27).
See the diagrams below to demonstrate these cube number examples.
 A cube with side length 2 units, volume 8 units ie 2 x 2 x 2 – we can also see there are 8 cubes.
2. A cube with side length 3 units, volume 27 units ie 3 x 3 x 3 – we can also see there are 27 cubes)
3 x 3 x 3 cube
When will children learn about cube numbers in primary school?
As part of the multiplication and division topic, the national curriculum for Key Stage 2 states that Year 5 pupils should be taught to:
 recognise and use square numbers and cube numbers, and the notation for squared (2) and cubed (3)
 solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes
In the nonstatutory notes and guidance, the curriculum advises that children understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 92 x 10).
This knowledge of square numbers will be built on in Year 6, particularly when learning about BIDMAS and the order of operations when children may learn the term ‘indices’ (an ‘index number’ is the name for the little 2 used to mean ‘squared’, or the little 3 used to mean ‘cubed’).
Cube roots, like square roots, or working with squares and cubes of decimals are not generally tackled by children until secondary school closer to GCSE.
How do cube numbers relate to other areas of maths?
Cube numbers are particularly useful when finding the volume of cubes, which children begin to do in Year 5 (pupils should be taught to estimate volume [for example, using 1cm3 blocks to build cuboids (including cubes)])
In Year 6, pupils should be taught to calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm3) and cubic metres (m3), and extending to other units [for example, mm3 and km3].
By Year 6 maths pupils will be taught to use their knowledge of the order of operations to carry out calculations and problem solving questions with cubed numbers, including twostep and multistep word problems.
Wondering how to explain other key numeracy vocabulary to your children? Check out our Maths Dictionary For Kids, or try the following explanations for parents of children following a maths mastery approach in their primary school:
 What Is Pemdas?
 What Is The Lowest Common Multiple?
 What Is The Highest Common Factor?
 What Are Prime Numbers?
 What Is Place Value?
Cube number
questions and answers
1. Order these from smallest to largest: 52 32 33 23
(Answer: 23 (8), 32 (9), 52 (25), 33 (27))
2. Which of these numbers are also square numbers? 13 23 33 43 53
(Answer: 13 (1), 43 (64))
3. Find two cube numbers that total 152.
(Answer: 125 (53) + 27 (33))
4. Write a number less than 100 in each space in this sorting diagram.
Answer:
Online 1to1 maths lessons trusted by schools and teachers
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What Are Cube Numbers? Definition, List, Solved Examples, Facts
What Are Cube Numbers?
We know that a “number squared” refers to the product of the number with itself. So, to find a square, we simply multiply the number with itself once. 3 = 27$
and so on.
Since a cube is a threedimensional shape with the same length, width, and height of its sides, its volume can be calculated by “cubing” its length.
A cube with sidelength 1 unit is called a unit cube. If “n” such unit cubes perfectly fit inside a given cube, then we say that the volume of a given cube is “n cubic units.”
Related Worksheets
List of Cube Numbers From 1 to 100
1  1  $1 \times 1 \times 1$  51  132651  $51 \times 51 \times 51$ 
2  8  $2\times2\times2$  52  140608  $52 \times 52 \times 52$ 
3  27  $3 \times 3 \times 3$  53  148877  $53 \times 53 \times 53$ 
4  64  $4 \times 4 \times 4$  54  157464  $54 \times 54 \times 54$ 
5  125  $5 \times 5 \times 5$  55  166375  $55 \times 55 \times 55$ 
6  216  $6 \times 6 \times 6$  56  175616  $56 \times 56 \times 56$ 
7  343  $7 \times 7 \times 7$  57  185193  $57 \times 57 \times 57$ 
8  512  $8 \times 8 \times 8$  58  195112  $58 \times 58 \times 58$ 
9  729  $9 \times 9 \times 9$  59  205379  $59 \times 59 \times 59$ 
10  1000  $10 \times 10 \times 10$  60  216000  $60 \times 60 \times 60$ 
11  1331  $11 \times 11 \times 11$  61  226981  $61 \times 61 \times 61$ 
12  1728  $12 \times 12 \times 12$  62  238328  $62 \times 62 \times 62$ 
13  2197  $13 \times 13 \times 13$  63  250047  $63 \times 63 \times 63$ 
14  2744  $14 \times 14 \times 14$  64  262144  $64 \times 64 \times 64$ 
15  3375  $15 \times 15 \times 15$  65  274625  $65 \times 65 \times 65$ 
16  4096  $16 \times 16 \times 16$  66  287496  $66 \times 66 \times 66$ 
17  4913  $17 \times 17 \times 17$  67  300763  $67 \times 67 \times 67$ 
18  5832  $18 \times 18 \times 18$  68  314432  $68 \times 68 \times 68$ 
19  6859  $19 \times 19 \times 19$  69  328509  $69 \times 69 \times 69$ 
20  8000  $20 \times 20 \times 20$  70  343000  $70 \times 70 \times 70$ 
21  9261  $21 \times 21 \times 21$  71  357911  $71 \times 71 \times 71$ 
22  10648  $22 \times 22 \times 22$  72  373248  $72 \times 72 \times 72$ 
23  12167  $23 \times 23 \times 23$  73  389017  $73 \times 73 \times 73$ 
24  13824  $24 \times 24 \times 24$  74  405224  $74 \times 74 \times 74$ 
25  15625  $25 \times 25 \times 25$  75  421875  $75 \times 75 \times 75$ 
26  17576  $26 \times 26 \times 26$  76  438976  $76 \times 76 \times 76$ 
27  19683  $27 \times 27 \times 27$  77  456533  $77 \times 77 \times 77$ 
28  21952  $28 \times 28 \times 28$  78  474552  $78 \times 78 \times 78$ 
29  24389  $29 \times 29 \times 29$  79  493039  $79 \times 79 \times 79$ 
30  27000  $30 \times 30 \times 30$  80  512000  $80 \times 80 \times 80$ 
31  29791  $31 \times 31 \times 31$  81  531441  $81 \times 81 \times 81$ 
32  32768  $32 \times 32 \times 32$  82  551368  $82 \times 82 \times 82$ 
33  35937  $33 \times 33 \times 33$  83  571787  $83 \times 83 \times 83$ 
34  39304  $34 \times 34 \times 34$  84  592704  $84 \times 84 \times 84$ 
35  42875  $35 \times 35 \times 35$  85  614125  $85 \times 85 \times 85$ 
36  46656  $36 \times 36 \times 36$  86  636056  $86 \times 86 \times 86$ 
37  50653  $37 \times 37 \times 37$  87  658503  $87 \times 87 \times 87$ 
38  54872  $38 \times 38 \times 38$  88  681472  $88 \times 88 \times 88$ 
39  59319  $39 \times 39 \times 39$  89  704969  $89 \times 89 \times 89$ 
40  64000  $40 \times 40 \times 40$  90  729000  $90 \times 90 \times 90$ 
41  68921  $41 \times 41 \times 41$  91  753571  $91 \times 91 \times 91$ 
42  74088  $42 \times 42 \times 42$  92  778688  $92 \times 92 \times 92$ 
43  79507  $43 \times 43 \times 43$  93  804357  $93 \times 93 \times 93$ 
44  85184  $44 \times 44 \times 44$  94  830584  $94 \times 94 \times 94$ 
45  91125  $45 \times 45 \times 45$  95  857375  $95 \times 95 \times 95$ 
46  97336  $46 \times 46 \times 46$  96  884736  $96 \times 96 \times 96$ 
47  103823  $47 \times 47 \times 47$  97  912673  $97 \times 97 \times 97$ 
48  110592  $48 \times 48 \times 48$  98  941192  $98 \times 98 \times 98$ 
49  117649  $49 \times 49 \times 49$  99  970299  $99 \times 99 \times 99$ 
50  125000  $50 \times 50 \times 50$  100  1000000  $100 \times 100 \times 100$ 
How to Cube Numbers
To cube numbers, simply multiply the square of a number by itself. 3 = 1$
Root (cubic, square) to the degree: solutions, tables, examples
Contents:

Degree with a natural indicator

Degree with integer exponent

cube root

root degree

Comparison of arithmetic roots

How to get rid of irrationality in the denominator

How to simplify irrational expressions using shortcut multiplication formulas
youtube.com/embed/BvMYQ5eCBGg» frameborder=»0″ allowfullscreen=»allowfullscreen»>
An expression of the form is called a power.
Here — base degrees, — exponent degrees.
back to contents ▴
Degree with natural exponent
The easiest way is to determine the degree with a natural (that is, whole positive) exponent.
By definition, .
The expressions «square» and «cube» have long been familiar to us.
To square a number is to multiply it by itself .
.
To cube a number is to multiply it by itself three times.
.
Raising a number to a natural power means multiplying it by itself times:
as well as a negative integer.
By definition,
.
This is true for . The expression 0 ^{ 0 } is not defined.
Let us also define what a degree with a negative integer exponent is.
Of course, all this is true for , since you cannot divide by zero.
For example,
The exponent can be not only an integer, but also fractional , that is, a rational number. In the article «Number Sets» we talked about what rational numbers are. These are numbers that can be written as a fraction, where is an integer and is a natural number.
Here we need a new concept — the root degree. Roots and degrees are two related topics. Let’s start with the already familiar arithmetic square root.
Definition.
The arithmetic square root of a number is a nonnegative number whose square is .
According to the definition,
In school mathematics, we only take the root of nonnegative numbers. The expression for us now makes sense only for .
The expression is always nonnegative, i.e. . For example, .
Properties of the arithmetic square root:
Remember the important rule:
By definition, .
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Cube root
Similarly, is the cube root of , which when raised to the third power gives .
For example, because ;
since ;
since .
Note that the third root can be taken from both positive and negative numbers.
Now we can define the th root of any integer.
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Root power
The root of a number is the number that, when raised to the th power, produces a number.
For example,
Note that the root of the third, fifth, ninth — in a word, any odd degree — can be extracted from both positive and negative numbers.
The square root, as well as the root of the fourth, tenth, in general, any even degree can be extracted only from nonnegative numbers.
So, is a number such that . It turns out that roots can be written as powers with a rational exponent. It’s comfortable.
By definition,
in general.
Let’s immediately agree that the base of the degree is greater than 0.
For example,
The expression is by definition equal to .
In this case, the condition that is greater than 0 is also satisfied.
For example,
Let’s remember the rules of action with degrees :
— when multiplying degrees, the indicators are added;
— when dividing the degree by the degree, the indicators are subtracted;
— when raising a degree to a power, the indicators are multiplied;
whether the root.
2.
3.
Here we have written the roots as powers and used the action formulas with powers.
4. Find the value of the expression at
Solution:
When we get
Answer: 0.5.
5. Find the value of the expression at
Solution:
With a = 12 we get
We have used the properties of degrees.
Answer: 144.
6. Find the value of the expression when b = — 5.
Solution:
With b = 5 we get:
Answer: 125.
7. Arrange in ascending order:
Solution:
Let’s write the expressions as powers with a positive exponent and compare.
Since then
Since then
Compare and evaluate their difference:
means
We get: therefore
Answer:
8. Express the expression as a power: 9 0003
Solution:
Let’s take out the degree with a smaller exponent:
Answer:
9. Simplify the expression:
Solution:
Let’s reduce bases 6 and 12 to bases 2 and 3:
900 28
(perform the division of degrees with the same bases)
Answer: 0.25.
10. What is the value of the expression for ?
Solution:
When we get
Answer: 9.
back to contents ▴
Comparison of arithmetic roots
11. Which of the numbers is greater: or?
Solution:
Let’s square both numbers (positive numbers):
Find the difference between the results:
since
So the first number is greater than the second.
Answer:
back to contents ▴
How to get rid of irrationality in the denominator
If a fraction of the form is given, then you need to multiply the numerator and denominator of the fraction by:
Then the denominator becomes rational.
If a fraction of the form or is given, then you need to multiply the numerator and denominator of the fraction by the conjugate expression to get the difference of squares in the denominator.
Conjugated expressions are expressions that differ only in signs. For example,
and and are conjugate expressions.
Example:
0003
Example 1
Example 2
Example 3
Example 4
If the sum of two roots is given in the denominator, then in the difference, write the one that is greater as the first number, and then the difference in the squares of the roots will be a positive number.
Example 5.
13. Compare and0003
that means
Answer: less.
back to contents ▴
How to simplify irrational expressions using reduced multiplication formulas
Let’s show some examples.
14. Simplify: expressions:
Example 5.
because
Example 6
Example 7
since
0003
Solution:
Equation:
Answer:
Answer: 1.
20. Calculate the value of the expression:
Solution:
Answer: 1.
21. Calculate the value of the expression: if
Solution.
If then then
Answer: — 1.
22. Calculate:
Solution:
Answer: 1.
Consider an equation of the form where
028 Details about such equations — in the article «Exponential Equations».
When solving equations of this type, we use the monotonicity of the exponential function.
23. Solve the equation:
a)
b)
c)
Solution.
23. Solve the equation:
Solution:
then
Answer: 1.
24. Solve the equation:
Solution:
Answer: 4.
25. Solve the equation:
Solution :
So
The answer is 0.2.
If you want to analyze more examples, sign up for online USE preparation courses in mathematics
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Publication updated:
07/07/2023
How to extract the cube root in Python ~ PythonRu
The cube root of a number is a value that, when cubed, gives the original number. In other words, the cube root is the value that, when multiplied three times, we can get the number under the root.
The cube root is represented by the symbol «3√». In the case of the square root, we used only the symbol ‘√’ without indicating the degree, which is also called the radical. 93.
Cube root in Python
To calculate the cube root in Python, use the simple math expression x ** (1. / 3.), which results in the cube root of x as a floating point value. To check whether the operation of extracting the root was performed correctly, round the result obtained to the nearest integer and raise it to the third power, then compare whether the result is equal to x.
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x=8 cube_root = x**(1. /3.) print(cube_root)Output
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2.0In Python, we use the ** operator to raise a number to a power. Specifying a power equal to 1/3 in an expression with ** allows you to get the cube root of a given number.
Extracting the cube root of a negative number in Python
We can’t find the cube root of negative numbers using the above method. For example, the cube root of the integer 64 should be 4, but Python returns 2+3.464101615137754j.
To find the cube root of a negative number in Python, you first need to use the abs() function, and then you can use the simple expression with ** presented earlier to calculate it.
Let’s write a complete function that will check if the input number is negative and then calculate its cube root accordingly.
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def get_cube_root(x): ifxOutput
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4 3. 9999999999999996As you can see, we need to round the result to get the integer value of the cube root.
Using the Numpy cbrt() function
The numpy library offers another way to find the cube root in Python, which is to use the cbrt() method. The np.cbrt() function calculates the cube root for each element of the array passed to it.
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import numpy as np cubes = [125, 64, 27, 8, 1] cube_roots = np.cbrt(cubes) print(cube_roots)Output
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[5.4. 3.2. 1.]The np.cbrt() function is the easiest way to get the cube root of a number. It does not experience problems with negative inputs and returns an integer, such as 4 for the number 64 passed as an argument, unlike the approaches described above.
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I created this blog in 2018 to share useful tutorials, documentation and tutorials in Russian.