Perfect Cube of Numbers: Formula, Cube Root & Properties

Perfect Cube is a number that is a product of three same integers. In other words, a Perfect Cube is a number equal to a number multiplied by itself thrice. If ‘N’ is a perfect cube and N= M³ it infers that N is a perfect cube of M, where N can be obtained by multiplying N with itself three times. The formula of the Perfect Cube is given by,

N = M³

where, N= Perfect Cube and M= cube root of N. 

For example, when we multiply 5 X 5 X 5, we get 125. Therefore 125 (= 5³) is a perfect cube. Interestingly, the cube of an even number is always even and the cube of an odd number is always odd. The major application of cubes may be seen in geometry where the volume of the three-dimensional structures is calculated.

Key Takeaways: Perfect Cube, Cube Root, Even number, Odd Number, Squares, Cubes.

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Perfect Cube

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Perfect Cube can be defined as the product of three same integers. It means a Perfect Cube is a number that is equal to a number multiplied by itself thrice. Let us assume ‘N’ is a perfect cube and N= M³. It implies that N is a perfect cube of M, where N can be obtained by multiplying N with itself three times. 

Perfect Cubes in Geometry

For example, 8, 27, 64 are some of the examples of perfect cubes as they can be obtained by multiplying 2,3 and 4 with themselves thrice. Perfect cubes can be negative as well. For example, -27 is a perfect cube of -3.

8 = 2X2X2 = 2³

27= 3 X 3X 3= 3³

64= 4X 4X4= 4³

-27= -3X-3X-3= (-3)³

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How to Find a Perfect Cube

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Perfect Cube of a number can be investigated by following the steps given below-

Step 1: Every Number can be expressed as the product of its prime factors. The given number has to be factorized starting from the smallest prime number which is 2.

Step 2: Once the prime factorization is done, every three same factors are to be clubbed together.

Step 3: Same steps should be repeated for all the sets of the group of the same three factors. Now, if there is any factor left behind and it is not being fitted into a group of the same factors, the given number is not a Perfect Cube. Otherwise, the said number is a perfect cube as every factor will fit into the group. Here, is an example to understand the concept-

Example 1: 216

Prime Factorization of 216 is 2X3 X 2X3 X 2X3

216= 6X6X6 (same number is multiplied thrice)

Thus, 216 is a Perfect Cube.

Example 2: 300

Prime Factorization of 300 is 2X5 X 2X5X3

300= 10X 10 X 3 (since the factors are not the same, 300 cannot be considered as a Perfect Cube.)

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Perfect Cube Formula

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Perfect Cube formula is used to see if any number is a perfect cube or otherwise. Let us assume that there is an integer N, such that, N=MXMXM. Now, as per the fundamental theorem of arithmetic, every composite number is expressed as the product of the power of its component prime factors. Here, if the power of all the prime factors is in multiple of three, then the number can be called a Perfect Cube.

N = M³

Where, 

N= Perfect Cube

M= cube root of N

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Properties of Perfect Cube

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Perfect Cubes have some fascinating properties that are listed below-

1. The cube of an even number is always even. For instance, 4 is an even number and 4³= 64 which is an even number as well.
2. The cube of an odd number is always odd. For instance, 5 is an even number and 5³= 125 which is an odd number as well.
3. Another interesting property of a Perfect Cube is that it 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.

Here, the total count of consecutive odd numbers which add up to constitute a perfect cube is always equal to the number that is being cubed. Thus, for 2³, it involves the sum of 2 numbers, 3 and 5 and for 3³, it involves three odd numbers 5,7 and 9. The pattern continues for the successive numbers as well.

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Cube Root of Perfect Cube

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Whenever a number is cubed, it implies that it has been multiplied by itself thrice. Cube root refers to reversing the process of cubing the number. For example, when 7 is cubed, we obtain 7X7X7 which is 343. Reversing the process, we will get the cube root of 343 which is 7. The symbol of cube root is  \(\sqrt[ 3]{ } \). It is the same as the use of the square root symbol except a ‘3’ is inserted in the place of 2. However, in the exponent form, the cube root of a number can also be expressed as (number)1/3.

Cube Root

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Things to Remember

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• A Perfect Cube can be defined as the product of three same integers. It means a perfect cube is a number that is equal to a number multiplied by itself thrice.
• If ‘N’ is a perfect cube and N= M³. It implies that N is a perfect cube of M, where N can be obtained by multiplying N with itself 3 times.
• The cube of an even number is always even.
• The cube of an odd number is always odd.
• A Perfect Cube is that it can be expressed as a sum of consecutive odd numbers.

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