Perfect Cube

A perfect cube is a number that can be expressed as the product of an integer multiplied by itself three times. This concept establishes a strong foundation for understanding higher mathematical principles and finds practical applications in various fields.

What are Perfect Cubes?

A perfect cube is a number that can be obtained by multiplying an integer (a whole number) by itself three times. In mathematical notation, a perfect cube can be represented as n3 where n is an integer. For example, 23 is a perfect cube because 2×2×2 equals 8. Similarly, 33 is also a perfect cube, yielding the value 27 (3×3×3).

We can find cubes of any number, whether integer or fraction. For example, the cube of 0.6 is:

0.6 × 0.6 × 0.6 = 0.216

How to Find the Perfect Cube?

How to determine if a number is a perfect cube by using prime factorization:

Step 1: Begin by finding the prime factorization of the given number, starting from the smallest prime number (2).

Step 2: Group the prime factors in sets of three, as each set of three factors will represent one cube.

Step 3: Repeat the step for all the group sets of the same three factors. 

If any prime factors are left ungrouped, the number is not a perfect cube. The number is a perfect cube if all factors are grouped in three sets.

Example 1

To prime factorize 216, we need to find its prime factors. Let’s go through the process:

Step 1: Divide by the smallest prime number, which is 2.

216=2×108

Step 2: Divide 108 by 2 again:

108=2×54

Step 3: Divide 54 by 2 once more:

54=2×27

Step 4: Now, let’s divide by the next prime number, which is 3:

27=3×9

Step 5: Finally, divide 9 by 3:

9=3×3

At this point, we’ve reached a prime factorization where all the factors are prime numbers:

216=23×33

So, the prime factorization of 216 is 23×33, which means 216 can be expressed as the product of three 2s and three 3s.

Therefore, 216 is a perfect cube.

Example 2

To prime factorize 200, we need to find its prime factors. Let’s go through the process:

Step 1: Divide by the smallest prime number, which is 2.

200=2×100

Step 2: Divide 100 by 2 again:

100=2×50

Step 3: Divide 50 by 2 once more:

50=2×25

Step 4: Now, let’s divide by another prime number, which is 5:

25=5×5

Step 5: Finally, divide 5 by 5:

5=5×1

The number 200 on prime factorization gives 2 × 2 × 2 × 5 × 5. Here, the prime factor 5 is not in the power of 3. Therefore, 200 is not a perfect cube.

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