What is a cube number: What are cube numbers?

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What 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 23 (“two cubed”).

Another example of a cube number is 27 because it’s 33 (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.

An expression of the form is called a power.

Here — base degrees, — exponent degrees.

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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.


The arithmetic square root of a number is a non-negative number whose square is .

According to the definition,

In school mathematics, we only take the root of non-negative numbers. The expression for us now makes sense only for .

The expression is always non-negative, 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 non-negative 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.



Here we have written the roots as powers and used the action formulas with powers.
4. Find the value of the expression at


When we get

Answer: -0.5.

5. Find the value of the expression at


With a = 12 we get

We have used the properties of degrees.

Answer: 144.

6. Find the value of the expression when b = — 5.


With b = -5 we get:

Answer: -125.

7. Arrange in ascending order:


Let’s write the expressions as powers with a positive exponent and compare.

Since then

Since then

Compare and evaluate their difference:


We get: therefore


8. Express the expression as a power: 9 0003


Let’s take out the degree with a smaller exponent:


9. Simplify the expression:


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 ?


When we get

Answer: 9.

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Comparison of arithmetic roots

11. Which of the numbers is greater: or?


Let’s square both numbers (positive numbers):

Find the difference between the results:


So the first number is greater than the second.


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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 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.

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How to simplify irrational expressions using reduced multiplication formulas

Let’s show some examples.

14. Simplify: expressions:

Example 5.


Example 6

Example 7






Answer: 1.

20. Calculate the value of the expression:


Answer: 1.

21. Calculate the value of the expression: if


If then then

Answer: — 1.

22. Calculate:


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:





23. Solve the equation:



Answer: -1.

24. Solve the equation:


Answer: 4.

25. Solve the equation:

Solution :


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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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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cube_root = x**(1. /3.)


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In 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):


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-3. 9999999999999996

As 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)


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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.


I created this blog in 2018 to share useful tutorials, documentation and tutorials in Russian.

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