A perfect cube is a number that is obtained by multiplying the same integer three times. For example, multiplying the number 4 three times results in 64. Therefore, 64 is a perfect cube. Therefore, perfect cube = number × number × number. The cube root of 64 is 4. A number is said to be a perfect cube if it can be decomposed into a product of the same three numbers.
Perfect Cube Definition
A perfect cube is defined as the product of three same integers. To check if a number 'N' is a perfect cube or not, we check whether an integer, when multiplied by itself thrice, gives the number 'N'. If yes, then it is a perfect cube. 1, 8, 27, 64 are some examples of perfect cubes. A perfect square is one that is obtained by multiplying a number by itself. It is different from a perfect cube. Perfect cubes can be both positive and negative numbers. For example, -64 is a perfect cube because it is the product of multiplying -4 three times.
Cube Root of Perfect Cubes
When we say that a number is cubed, it means it is multiplied three times. Cube root is reversing the process of cubing a number. For example, when the number 5 is cubed, we get 5 × 5 × 5, which is 125. The cube root of 125 is 5. This is because we get 125 on multiplying the number 5 three times. The symbol for cube root is ∛. It is the same as the square root symbol except that a '3' is inserted in it, to show that it denotes a cube root. In the exponent form, the cube root of a number can be expressed as [number]1/3.
How to Find the Perfect Cube?
The perfect cube of a number can be checked by following the steps given below:
- Step 1: Prime factorize the given number starting from the smallest prime number [2].
- Step 2: Once the prime factorization is done, club every three same factors together.
- Step 3: Repeat the step for all the sets of the group of the same three factors. If there are any factors that are left behind and do not fit into a group of three same factors, then the given number is not a perfect cube. Otherwise, the given number is a perfect cube.
Let us understand this with an example.
There is a trick to check if larger numbers are a perfect cube or not. To check that, find the sum of all the digits of the number repeatedly and see if it is either 0, 1, 8, or 9. If it is any one of these, then it MAY BE a perfect cube, but not necessarily true. To confirm whether it is a perfect cube we will have to again follow the same process of prime factorization method discussed above.
Note: This is a trick to check if a number may or may not be a perfect cube but it is not always guaranteed to be a perfect cube.
Example: The reason why this trick is used is that suppose we are given a number, say, 8768, and we are not sure if this is a perfect cube or not. So, we find the sum of the digits of this number and check if it is either 0, 1, 8, or 9. If it is not either of these four numbers, then we need not factorize the number to check if it is a perfect cube. In other words, this trick just eliminates the process of prime factorization if the sum is not equal to 0, 1, 8, or 9. In this case, the sum of the digits of the numbers 8768 is 29 and the sum of 2 and 9 is 11 and finally, 1 + 1 is equal to 2. Therefore, it is not 0, 1, 8, or 9. So we can easily conclude that 8768 is not a perfect cube.
There is a special case wherein we get the sum of the digits equal to either 0, 1, 8, or 9 but the number is still not a perfect cube. For example, in the number 3456, the sum of the digits in 3 + 4 + 5 + 6, which is 18 and then 1 + 8 = 9. Though the sum of the digits is 9, when we factorize the number we get, 3456 = 2 × 3 × 2 × 3 × 2 × 3 × 2 × 2 × 2 × 2. So we get, 6 × 6 × 6 × 4 × 4. Since we are short of one 4 to make a group of three same numbers, we can say that 3456 is not a perfect cube.
Perfect Cube Formula
The perfect cube formula is used to check if any number is a perfect cube or not. Suppose we have a number, x, such that, x = y × y × y. As per the fundamental theorem of arithmetic, every composite number can be expressed as the product of the power of its prime factors. If the power of all the prime factors is in the multiple of 3, then the number is said to be a perfect cube.
List of Perfect Cubes from 1 to 50
The perfect cubes of numbers from 1 to 50 are listed in the table given below. The perfect cubes are obtained by multiplying every number thrice by itself.
| 1 | 1 × 1 × 1 | 1 |
| 2 | 2 × 2 × 2 | 8 |
| 3 | 3 × 3 × 3 | 27 |
| 4 | 4 × 4 × 4 | 64 |
| 5 | 5 × 5 × 5 | 125 |
| 6 | 6 × 6 × 6 | 216 |
| 7 | 7 × 7 × 7 | 343 |
| 8 | 8 × 8 × 8 | 512 |
| 9 | 9 × 9 × 9 | 729 |
| 10 | 10 × 10 × 10 | 1000 |
| 11 | 11 × 11 × 11 | 1331 |
| 12 | 12 × 12 × 12 | 1728 |
| 13 | 13 × 13 × 13 | 2197 |
| 14 | 14 × 14 × 14 | 2744 |
| 15 | 15 × 15 × 15 | 3375 |
| 16 | 16 × 16 × 16 | 4096 |
| 17 | 17 × 17 × 17 | 4913 |
| 18 | 18 × 18 × 18 | 5832 |
| 19 | 19 × 19 × 19 | 6859 |
| 20 | 20 × 20 × 20 | 8000 |
| 21 | 21 × 21 × 21 | 9261 |
| 22 | 22 × 22 × 22 | 10648 |
| 23 | 23 × 23 × 23 | 12167 |
| 24 | 24 × 24 × 24 | 13824 |
| 25 | 25 × 25 × 25 | 15625 |
| 26 | 26 × 26 × 26 | 17576 |
| 27 | 27 × 27 × 27 | 19683 |
| 28 | 28 × 28 × 28 | 21952 |
| 29 | 29 × 29 × 29 | 24389 |
| 30 | 30 × 30 × 30 | 27000 |
| 31 | 31 × 31 × 31 | 29791 |
| 32 | 32 × 32 × 32 | 32768 |
| 33 | 33 × 33 × 33 | 35937 |
| 34 | 34 × 34 × 34 | 39304 |
| 35 | 35 × 35 × 35 | 42875 |
| 36 | 36 × 36 × 36 | 46656 |
| 37 | 37 × 37 × 37 | 50653 |
| 38 | 38 × 38 × 38 | 54872 |
| 39 | 39 × 39 × 39 | 59319 |
| 40 | 40 × 40 × 40 | 64000 |
| 41 | 41 × 41 × 41 | 68921 |
| 42 | 42 × 42 × 42 | 74088 |
| 43 | 43 × 43 × 43 | 79507 |
| 44 | 44 × 44 × 44 | 85184 |
| 45 | 45 × 45 × 45 | 91125 |
| 46 | 46 × 46 × 46 | 97336 |
| 47 | 47 × 47 × 47 | 103823 |
| 48 | 48 × 48 × 48 | 110592 |
| 49 | 49 × 49 × 49 | 117649 |
| 50 | 50 × 50 × 50 | 125000 |
Properties of Perfect Cubes
Perfect cubes have some interesting properties which are listed below.
- The cube of an even number is even. For example, 2 is an even number, and 23 = 8.
- The cube of an odd number is odd. For example, 3 is an odd number, and 33 = 27.
- Perfect cube of a number can be expressed as a sum of consecutive odd numbers. For example, 13 = 1 , 23 = 3 + 5. 33 = 5 + 7 + 9, 43 = 13 + 15 + 17 + 19 and so on.
- The total count of consecutive odd numbers which sum up to form a perfect cube is equal to the number that is being cubed. For example, to get 13, only 1 odd number is involved, for 23, it involves the sum of 2 numbers, 3 + 5. For 33, it is 5 + 7 + 9, where three odd numbers are involved. To get 43, we add 13 + 15 + 17 + 19. The same pattern continues for the successive numbers also.
Topics Related to Perfect Cube
Check out some interesting topics related to a perfect cube.
- Cube Root
- Cube 1 to 10
- Exponents Squares and Cubes
- Perfect Square
-
Example 1: Use the perfect cube formula to find if 729 is a perfect cube.
Solution: The perfect cube formula is m = ∛N, where 'N' is the perfect cube, and 'm' is the cube root of 'N'.
Given, N = 729, the cube root of 729 = ∛729.
Prime factorization of 729 = 3 × 3 × 3 × 3 × 3 × 3. Now, we club each set of 3 factors, which are the same. Therefore, we get [3 × 3] × [3 × 3] × [3 × 3].
or 729 = [3 × 3]3 or 93
Therefore, cube root of 729 = 9Hence, 729 is a perfect cube.
Example 2: Find the cube root of 512 and check if it is a perfect cube or not.
Solution:
Let us find the cube root of 512.
Prime factorization of 512 = 8 × 8 × 8 or 83 Therefore, ∛512 = 8.
Since there are exactly three factors that are the same, we can say that 512 is a perfect cube.
Example 3: If the value of y3 = 1000, find the value of 'y' using the perfect cube formula.
Solution:
The perfect cube formula is m = ∛N, where 'N' is the perfect cube, and 'm' is the cube root of 'N'.
Given, y3 = 1000, therefore, y = ∛1000. Prime factorization of 1000 = 2 × 2 × 2 × 5 × 5 × 5
or 1000 = [2 × 5]3 or 103
Therefore 103 = 1000. Hence y = 10.
So, the value of y = 10.
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FAQs on Perfect Cube
A perfect cube is a number that is obtained by the multiplication of the same number three times. For example, when we multiply 7 × 7 × 7, we get 343. Therefore, 343 is a perfect cube.
How to Find if a Number is a Perfect Cube or Not?
To find if a number is a perfect cube or not, the following steps need to be followed:
- Do the prime factorization of the number and find its factors.
- Club or group a set of three same factors together.
- If there are no factors left ungrouped, then the number is a perfect cube.
- For example, the prime factorization of 8 is 2 × 2 × 2. Therefore, there are no factors that are left ungrouped. Hence, 8 is a perfect cube.
What is the Difference Between a Cube Root and a Perfect Cube?
A cube root is denoted by the symbol ∛ or [number]1/3. For example, the cube root of 125 is 5. This is expressed as ∛125 = 5. To check if a number is a perfect cube or not, we find the cube root of the given number. The cube root should be a whole number which will prove that the given number is a perfect cube. For example, to check if 27 is a perfect cube, we find the cube root of 27, which is denoted as ∛27 or [27]1/3 and we get the result as 3. Since 3 is a whole number, this means that 27 is a perfect cube.
Can Perfect Cube Numbers Be Negative?
Yes, perfect cube numbers can be both positive and negative. For example, -125 is also a perfect cube since it is obtained by multiplying -5 three times. In other words, -5 × -5 × -5 = -125. Therefore, negative numbers can also be perfect cube numbers.
What is the Perfect Cube Formula?
Let us consider a number 'm', which when multiplied thrice yields 'N'. To check if it is a perfect cube or not, we can use the following formula: Given, N = m × m × m.
N = m3
m = ∛N
If the condition m = ∛N, is true, then the number 'N' is a perfect cube. For example, 64 = 4 × 4 × 4 or 64 = 43. Therefore, ∛64 = 4. Hence, we conclude that 64 is a perfect cube.
What is the List of Perfect Cubes from Numbers 1 to 10?
Perfect cube numbers can be obtained by multiplying every number thrice by itself. For example, 1 × 1 × 1 = 1 and 2 × 2 × 2 = 8 and so on. The list of perfect cubes from 1 to 10 is as follows: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.
Are Perfect Cube Numbers Even or Odd?
The perfect cube of an even number is always even and the perfect cube of an odd number is always odd. For example, the perfect cube of 6 is 216 and the perfect cube of 7 is 343.
Is there Any Shortcut Way to Find if a Large Number is a Perfect Cube or not?
To check if a large number is a perfect cube or not, we find the sum of the digits of the given number, which means add the individual digits of the given number and check if the sum equals 0, 1, 8, or 9. If it is true, then the number MAY BE a perfect cube but not necessarily true. To make sure if the number is a perfect cube, we follow the prime factorization method that we do for smaller numbers. This trick just eliminates the process of prime factorization if the sum is not equal to 0, 1, 8, or 9.
What is the Perfect Cube Formula for Polynomials?
In the case of polynomials, we can factor polynomials by applying algebraic identities such as,
- Factoring a sum of cubes: a3 + b3 = [a + b][a2 – ab + b2]
- Factoring a difference of cubes: a3 – b3 = [a – b][a2 + ab + b2]
How to Use Perfect Cube Formula?
We know that any number can be expressed as the product of the power of its prime factors.
- Step 1: Write the number as the product of its prime factors.
- Step 2: Check if the power of all the prime factors of the number is in the multiple of 3.
Then the number is said to be a perfect cube.
What is the Perfect Cube Formula for Negative Numbers?
In the case of negative numbers, the perfect cube formula is the same. It's just that the cube of a negative integer is always negative such as [−4] × [−4] × [−4] = −64.