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Cubes and Cube Roots

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Cube and Cube Roots are fundamental concepts in algebra and it is introduced in the early classes. The multiplication of a number to itself gave rise to a square and then if we multiply the number by its square then the result becomes a cube and the inverse of the cube is the cube root which we will study in this article. In this article, we will learn about cubes and cubes and also learn about the methods to find both cubes of a number and cube roots of a number. So, let’s dive into the concept of Cubes and Cube Roots.

What is Cube?

Physically a cube is a solid figure which has all its sides equal. In mathematical terms, the cube of a number is the result of multiplying a whole number by itself twice, i.e., the whole number is used three times just like the sides of a cube. For example: 

  • Cube of 1 = 13 = 1 × 1 × 1 = 1,
  • Cube of 2 = 23 = 2 × 2 × 2 = 8,
  • Cube of 3 = 33 = 3 × 3 × 3 = 27 . . . and so on.

Cube 1 to 20

The cube of natural numbers 1 to 20 is discussed in the table below,

Cube 1 to 20

 

Properties of Cube of Numbers

There are various properties of the cube of numbers, some of which are as follows:

  • Cubes of all odd numbers are odd.
  • Cubes of all even numbers are even.
  • Cubes of numbers ending with 2 will end in 8. Similarly, cubes of numbers ending with 8 always end with 2.
  • Cubes of numbers ending with 3 will end in 7. Similarly, cubes of numbers ending with 7 always end with 3.
  • Cubes of numbers ending with 4, 5, 6, and 9 will end in 4, 5, 6, and 9 respectively.

Representation of Cube Numbers

Cube Numbers can be represented as the volume of cubes as volume is also a three-dimensional quantity and calculated by the multiplication of side length by itself three times. The following illustration shows the representation of cube numbers:

Representation of Cube Numbers

 

Cube of Negative Numbers

Similar to positive integers we can find the cube of negative integers as well. It is the same for negative numbers as well. To find the cube of a negative number we just multiply the negative number three times by itself. As a product of two negative numbers is positive and the product of negative and positive numbers is negative. Thus, a cube of a negative number is always negative. 

For example, a cube of -2 is nothing but (-2)×(-2)×(-2) = 4×(-2) = -8.

  • Cube of (-3) = -27
  • Cube of (-4) = -64
  • Cube of (-5) = -125 . . . and so on,

Cube of Fractions

A cube of any number is defined as the product of itself three times or the number raised to the third power. For fractions, cubes can be calculated by individually calculating the cubes of the numerator and denominator in the fraction. For example, a cube of (3/2) is simply 33/23 = 27/8 [As the cube of 3 is 27 and the cube of 2 is 8]. Some more examples of cubes of fractions are as follows:

  • Cube of (-1/3) = -1/27
  • Cube of (4/5) = 64/125
  • Cube of (7/5) = 343/125

Unit Digits in Cube Numbers

As we discussed in the properties of the cube 

  • If a number is odd, its cube number unit digit is also odd.
  • And similarly, If a number is even, its cube number unit digit is also even.

Following is a table that shows the unit digit of a number and the unit digit of the cube of that number.

Units Digit of Number

Units Digit of its Cube

1

1

2

8

3

7

4

4

5

5

6

6

7

3

8

2

9

9

Perfect Cubes

A perfect cube is a positive integer equal to some other positive integer raised to the third power. In other words, when we multiply an integer by itself the resulting number is called a perfect cube. For example, 125 is a perfect cube since 125 = 5 × 5 × 5 = 53. However, 121 is not a perfect cube because there is no integer n such that n3 = 121 = 112. Some other examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, . . . , etc.

What is Cube Root?

The cube root of a number is a value that, when multiplied three times, gives that number. For example, 3 × 3 × 3 = 27, so the cube root of 27 is 3.

Symbol of Cube Root

The cube root of any number in mathematics is represented as \bold{\sqrt[3]{x}} . Some examples of the use of symbols are as follows:

  • \sqrt[3]{27} = 3
  • \sqrt[3]{125} = 5
  • \sqrt[3]{\left(\frac{1}{27}\right)} = \frac{1}{3}

Cube Root Formula

The cube root formula is the formula that is used to find the cube root of any substance. Suppose the cube of a number is x is y we can represent this as,

x3 = y

Now the cube root of y is calculated as,

y1/3 = x

The image added below represent the same.

Cube Root Formula

 

How to find the Cube Root of a Number?

We can easily find the cube root of a number using the following methods,

  • By Prime Factorization
  • By Estimation Method

Now, let’s learn about them in detail.

By Prime Factorization

Prime factorization is a method through which you can easily determine whether a particular figure represents a perfect cube. If each prime factor can be clubbed together in groups of three, then the number is a perfect cube. 

For example: Let us consider the number 1728. 

As prime factorization of 1728 is 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be rewritten as 23 × 23 × 33

Here, all the numbers can be clubbed into groups of three. So we can definitely say that 1728 is a perfect cube. In fact, the cube root of 1728 is a product of numbers taken one from each group i.e.,  2 × 2 × 3 = 12. 

Example 1: Find the cube root of 216.

Answer:

216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 * 33 = 63

Hence, the cube root of 216 is 6.

Example 2: Find the cube root of 5832.

Answer:

5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 

5832 = 23 × 33 × 33 = 183

Hence, the cube root of 5832 is 18.

Cube Root of a Cube Number using Estimation

In this method, we can estimate the cube of a perfect cube number using some rule. The steps to estimate the cube of perfect cube number are as follows:

Step 1: Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number. 

So 117649 has two groups, and the first group is (649) and the second group is (117).

Step 2: The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since 649 ends with 9, the cube root’s unit digit is 9.

Note: We can use the following table for finding the unit digit of cube root,

Unit digit of Cube Root 1 2 3 4 5 6 7 8 9
Unit digit of its Cube 1 8 7 4 5 6 3 2 9

Step 3: Find the cube of numbers between which the second group lies. The other group is 117. 

We know that 403= 64000 i.e., second group for cube of 40 is 64, and 503= 125000 i.e., second group for 50 is 125. As 64 < 117 < 125. Thus, the ten’s digit of the requred number is either 4 or 5 and 50 is the least number with 5 as ten’s digit. Thus, 4 is the ten’s digit of the given number.

So, 49 is the cube root of 117649.

Note: For help with second group we can use the following table,

Number 0 10 20 30 40 50 60 70 80 90
Cube 0 1000 8000 27000 64000 125000 216000 343000 512000 729000

Example: Estimate the Cube root of the number 357911.

Solution:

Let’s take another cube number, say 175616.

Step 1: Starting from the rightmost digit, group the digits in threes. So, the first group is (616) and the second group is (175).

Step 2: The unit digit of the first group (616) is 6, which corresponds to the unit digit of the cube root 6.

Step 3: Find the cubes of numbers between which the second group lies. We know that 43³ = 79507 and 44³ = 85184. Since 175 is between 79507 and 85184, the tens digit of the required number is 4.

Therefore, the cube root of 175616 is 46.

Hardy-Ramanujan Numbers

Numbers like 1729, 4104,13832, etc. are known as Hardy – Ramanujan Numbers because they can be expressed as the sum of two cubes in two different ways. A number n is said to be Hardy-Ramanujan Number if

n = a3 + b3 = c3 + d3

Where,

  • a, b, c, and d are all distinct positive integers

The first four Hardy-Ramanujan numbers are:

  • 1729 = 13 + 123 = 93 + 103
  • 4104 = 23 + 163 = 93 + 153
  • 13832 = 23 + 243 = 183 + 203
  • 20683 = 103 + 273 = 193 + 243

If we consider negative integers as well, then 91 becomes the smallest Hardy-Ramanujan Number as it can be expressed as follows:

91 = 63 + (-5)3 = 43 + 33

Note: The number 1729 is also sometimes referred to as the taxicab number as it was the number of the taxi taken by Dr. Hardy while going to meet Ramanujan at the hospital.

FAQs on Cube and Cube Roots

Q1: What is the difference between 3D Cube and Cube Numbers?

Answer:

3D Cube is solid and a physical object as cube numbers are the abstract concept which is a number we get after raising any number to it’s third power.

Q2: What is the Cube of 3?

Answer:

Cube of 3 is 27.

Q3: What is the Cube of 4?

Answer:

Cube of 4 is 64.

Q4: How do you calculate the cube root of a number?

Answer:

We can find the cube root of a perfect cube number using the prime factorization method or we can estimate the cube root using the steps explained in this same article.

Q5: What is the Cube Root of 27?

Answer:

A cube of 3 is 27, therefore cube root of 27 is 3.

Q6: What is the Cube Root of 125?

Answer:

A cube of 5 is 125, therefore cube root of 125 is 5.

Q7: What is the relationship between Cubes and Cube roots?

Answer:

Cube and Cube Roots are inverse operations that undo the effect of each other on any number. In other words, if we take any number and take it’s third power to find its cube and then find the cube root of the result. We will end up with the same number we have taken at first.


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Last Updated : 31 May, 2023
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