Perfect Cube Calculator

What is the perfect cube

In math, a perfect cube represents a number that is made by multiplying a certain integer (whole number) three times by itself. For example, if we multiply the number 6 three times, we get:

7 \cdot 7 \cdot 7 = 343

343 is a perfect cube, because we can get it by calculating an integer, in this case 7, by itself three times.

Which numbers are perfect cubes?

A perfect cube is a product of three same integers – numbers with no decimal or fractional part. They can be both positive and negative.

If we want to check if a certain number ”X” is a perfect cube or not, we will first see if there is an integer that, when multiplied three times by itself, gives us “X”. If an integer like that exists, then “X” is a perfect cube.

We have to make a distinction between a perfect cube and a perfect square. The perfect square can be calculated by squaring an integer (multiplying it by itself):

PS=x^2 = x \cdot x \text { (x is an integer)}

A perfect cube is, as we mentioned before, calculated by cubing an integer:

PC=x^3=x \cdot x \cdot x \text { (x is an integer)}

Perfect cube formula

The simplest way to show the perfect cube is by using a simple formula:

\text {PC} = x \cdot x \cdot x

This can also be expressed as:

PC = x^3

We can say that a certain number is a perfect cube if it can be decomposed by the same number 3 times. 

Opposite the perfect cube, we will find the cube root. It represents a reversed process of cubing a number. The cube root of 343 is 7.

\sqrt [3]{343} = 7

As we can see, the symbol for the cube root is ∛. It is similar to the square root which we are all familiar with. The 3 tells us that we are looking for the 3rd (cube) root. So, if that number was 5, we would be looking for the 5th root. If there is no number, it is implied that a square root is needed.

Mathematically, the cube root of a number can also be expressed as:

x= y^{\frac {1}{3}}

Can the cube root of a number be negative?

A cube root can be any real number – positive, negative, or even a zero. The result of a negative number cube will always be smaller than zero.

(-6)³= -216

You can find more examples of this in the table of perfect cubes below.

The perfect square, on the other hand, can only be a positive number, as when you multiply two negative numbers, you get a positive number. In other words, it is impossible to multiply an integer by itself, and have the product be smaller than zero.

List of perfect cubes

Number	Perfect Cube	Number	Perfect Cube
1	1	-1	-1
2	8	-2	-8
3	27	-3	-27
4	64	-4	-64
5	125	-5	-125
6	216	-6	-216
7	343	-7	-343
8	512	-8	-512
9	729	-9	-729
10	1000	-10	-1000
11	1331	-11	-1331
12	1728	-12	-1728
13	2197	-13	-2197
14	2744	-14	-2744
15	3375	-15	-3375
16	4096	-16	-4096
17	4913	-17	-4913
18	5832	-18	-5832
19	6859	-19	-6859
20	8000	-20	-8000

Looking at the previous chart, we can easily find examples of perfect cubes. As we can see, 27 is a perfect cube because we can get it by multiplying the number 3 three times.

Difference of perfect cubes

The difference of two perfect cubes equals the difference of their cube roots multiplied by the sum of their squares and the product of their cube roots. Mathematically:

a^3-b^3= (a-b)(a^2+ab+b^2)

As we can see, this formula contains both binomial (a-b) and trinomial (a²+ab+b²) factors.

The binomial factor is made up of the two cube roots of the perfect cubes.

(a-b)

The trinomial factor is made up of the squares of the cube roots and the product of the cube roots. 

a^2+ab+b^2

We will explain this formula with a real example. Let’s find the difference between the perfect cubes of 216, whose cube root is 6, and 125, whose cube root is 5. This means that 6 is a and 5 is b.

If we substitute the values in the equation, we will get:

216 – 125 = (6 – 5)(36 + 30 + 25); \\ 6 – 5 = 1; \\ 36 + 30 + 25 = 91; \\ 1 \cdot 91 = 91

Now let’s see if this is true:

216-125=91

How do we use the perfect cube calculator? 

Calculating the perfect cube using the CalCon Perfect Cube Calculator is very simple because all you need to do is:

1. Enter the number from which you want to calculate the perfect cube
2. Enter the degree of the exponent (number 3) in the field
3. Read the result

FAQ