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# 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.

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## 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.

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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.

## FAQs

#### What is a perfect cube?

A perfect cube is a number that can be expressed as the product of an integer multiplied by itself three times. In mathematical notation, it is represented as n3, where n is a whole number.

#### How do I determine if a number is a perfect cube?

You can use methods like prime factorization or checking the sum of digits to determine if a number is a perfect cube. If you can find an integer whose cube equals the given number, then the number is a perfect cube.

#### Can a negative number be a perfect cube?

Yes, negative numbers can be perfect cubes. For instance, -27 is a perfect cube because (−3)3=−27.

#### What is the connection between perfect cubes and volume?

The volume of a cube with side length n is n3, the same as the cube of n. So, perfect cubes are related to the concept of three-dimensional volume.

#### How are perfect cubes used in real life?

Perfect cubes have engineering, architecture, and computer graphics applications. They help calculate volumes, surface areas, and measurements in three-dimensional objects.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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