Cube Root

Whenever a number (x) is multiplied three times, then the resultant number is known as the cube of that number. Thus, the cube for the number (x) becomes x³ or x-cubed. For example, let us take the number 5. We know that 5 × 5 × 5 = 125. Hence, 125 is called the cube of 5. While on the other hand, the cube root of a number is the reverse process of the cube of a number and is denoted by ∛. Considering the same example, 5 is called the cube root of the number 125. On this page, we will learn more about the cubes and cube roots of a number.

Cube Root Definition

When we think about the words cube and root, the first picture that might come to our mind is a literal cube and the roots of a tree. Isn’t it? Well, the idea is similar. Root means the primary source or origin. So, we just need to think “cube of which number should be taken to get the given number”. In mathematics, the definition of cube root is written as Cube root is the number that needs to be multiplied three times to get the original number. Now, let us look at the cube root formula, where y is the cube root of x. ∛x = y. The radical sign ∛ is used as a cube root symbol for any number with a small 3 written on the top left of the sign. Another way to denote cube root is to write 1/3 as the exponent of a number.

Cube root is an inverse operation of the cube of a number.

Perfect Cubes

A perfect cube is an integer that can be expressed as the product of three same or equal integers. For example, 125 is a perfect cube because 5³ = 5 × 5 × 5 = 125. However, 121 is not a perfect cube because there is no number, which when multiplied three times gives the product 121. In other words, a perfect cube is a number whose cube root is an integer. The following table shows the perfect cubes of the first 10 natural numbers.

Number/Cube root	Perfect cube
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000

How to Find Cube Root?

The cube root of a number can be determined by using the prime factorization method. In order to find the cube root of a number, start with the prime factorization of the given number. Then, divide the factors obtained into groups containing three same factors. After that, remove the cube root symbol and multiply the factors to get the answer. If there is any factor left that cannot be divided equally into groups of three, that means the given number is not a perfect cube and we cannot find the cube root of that number. For example, let us see how we find the cube root of 15625.

Cube Root Formula

The cube root formula helps in the calculation of the cube root of any given number that is expressed in a radical form using the symbol ∛. It can be calculated by first finding out the prime factorization of the given number and then later applying the cube root formula. Suppose, x is any number such that, x = y × y × y.

The formula to calculate the cube root is given as:

Cube root of x = ∛x = ∛(y × y × y) = y

where,

y is the cube root of any number x.

This also means that the number x would be a perfect cube if y has an integer value.

Applications of Cube Root Formula

Given below are a few major applications of cube root formula,

• solve cubic equations.
• find the dimensions of a cube if the volume is given.
• provide a more precise dimension of the apartment.

What is Cube of a Number?

When we multiply a number three times by itself, the resultant number (product) is known as the cube of the original number. We call it a cube because it is used to represent the volume of a cube. In other words, a number raised to exponent 3 is called the cube of that number. For example, the cube of 3 is 27. That means 3 × 3 × 3 = 27, and it can be written as 3³. Similarly, the cube of 4 is 64, and the cube of 5 is 125, and so on.

To find the cube of a number, first, multiply that number by itself, then multiply the product obtained with the original number again. Let us find the cube of 7 through the same process. We know that the cube of a number N is N × N × N. So, the cube of 7 is 7 × 7 × 7. Now, in order to find the cube of 7, we will first find the value of 7 × 7. This value is 49. Now, we will find 49 × 7. This is equal to 343. Hence, we can say that cube of the number 7 is 343.

Cube of a fraction

Similar to the cube of a number, the cube of a fraction can be found by multiplying it three times. For example, the cube of the fraction (2/5) can be written as 2/5 × 2/5 × 2/5. Simplifying it further, we get the value of cube as (2 × 2 × 2) / (5 × 5 × 5). This is equal to (2³/ 5³) = 8/125.

Cube of Negative Numbers

The process to find the cube of a negative number is the same as that of a whole number and fraction. Here, always remember that the cube of a negative number is always negative, while the cube of a positive number is always positive. For example, let us try finding the cube of -7. We know that the cube of -7 is (-7) × (-7) × (-7). Now, in order to find the cube of (-7), we will first find the value of (-7) × (-7). This value is 49. Now, we will find 49 × (-7). This is equal to -343. Hence, we can say that cube of the number -7 is -343. The cube root formula for negative numbers is:

• Perform the prime factorization method of the given number.
• Divide the factors obtained into three groups, each of which should contain the same number of each factor.
• Multiply the factors in any one group to get the cube root.
• It’s just that the product of three negative values gives us a negative result. which is represented by the negative sign with the cube root of a negative number.

Cube Root Examples

What is Cube Root Used For?

How Do You Simplify a Cube Root?

Is it Possible to Simplify the Negative Cube Root?

What is Not a Perfect Cube?

What is the Cube of an Odd Natural Number?

Can the Cube Root of Any Odd Number be Even?

What is an Easy Way to Calculate the Cube Root of Any Number?

What is the Cube Root Formula in Algebra?

What is the Cube Root Formula for Negative Numbers?

How to Use Cube Root Formula?

What Is the Cube Root Formula in Algebra?

In math, the cube root formula is used to represent any number in the form of its cube root, such as for any number x, its cube root will be ³√x = x^1/3. For example, the cube root of 125 is 5 because 5 × 5 × 5 = 125.

What Is the Cube Root Formula for Negative Numbers?

Yes, it is possible to find the cube root of negative numbers. For example, -64 = (-4) × (-4) × (-4). We can write -64 as the product of 3 negative 4’s. Thus, ∛-64 = -4 because the product of three negative values gives us a negative result.

How To Use Cube Root Formula?

In order to use the cube root formula

• Step 1: Determine the prime factors of the number, say x using the prime factorization method.
• Step 3: Make three groups of factors so obtained, containing the same number of each factor.
• Step 3: Write that in the form of  ³√x = ³√(y×y×y), where y corresponds to the cube root of x.

How To Write Cube Root Formula in Words?

The cube root of any number is the number raised to the power of 1/3. 

Question 1: How to Find the Cube Root of an Integer?

Answer: The easiest way of how to find the cube root of a number is:

Make an estimate for the cube root. Never mind, if this value is not close to the real root.

Example. To find the cube root of 756, divide the number by the estimate twice and take the average of estimate, estimate, and the remainder.

756/20 75.6

75.6/10=7.56

Average of 10,10,7.56 = 27.56/3 = 9.18.

But this value is not very close to the initial estimate.

So, we need to repeat the above process taking the estimate as 9.

(We need to start with a round figure such that divisions can be done easily, just like given below)

756/9 = 84

84/9 = 9.33

The Average of 9,9 and 9.33 is equal to 27.33/3 = 9.11

Question 2: How do you Find the Cube Root of 2197?

Answer: 2197 is said to be a perfect cube because 13 x 13 x 13 is equal to 2197. Since the number 2197 is a whole number, we can say that it is a perfect cube. The nearest previous perfect cube of the number is 1728 and the nearest next perfect cube of the number is 2744. 2197 is known to be a perfect cube because 13 x 13 x 13 is equal to 2197.

Question 3: What Comes After the Cube Root?

Answer)The number of times the radicand is multiplied by itself. 2 means square root, 3 means cube root. After that, they are known as the 4th root, 5th root, and so on. If this is missing, it is assumed to be 2 – the square root.

Cube Root Formula for Negative Numbers

To find the cube root of any number, the prime factorization method is the best way out.

• In the case of negative numbers as well, perform the prime factorization of the given number.
• Divide the factors obtained into three groups, each of which should contain the same number of each factor.
• Multiply the factors in any one group to get the cube root.
• It’s just that the product of three negative values gives us a negative result. which is represented by the negative sign with the cube root of a negative number.

Cube Root Formula

The cube root formula is the inverse of the cube formula. In the cube formula, we multiply a number three times to get its cube, so in this case, we break down a number to be expressed as a product of three equal number and thus we get the cube root.

Let us look into this with an example: What is the cube root of 1331?

• The factors of 1728 are given as,
• 1728 = 11 × 11 × 11
• ³√1331 = ³√(11 × 11 × 11)
• ³√1331 = 11

Another way to calculate the cube root is as follows using the below table is as follows:

13 = 1	If the last digit of perfect cube number =1, the last digit of cube root for that number=1
23 = 8	If the last digit of perfect cube number =8, the last digit of cube root for that number=2
33 = 27	If the last digit of perfect cube number =7, the last digit of cube root for that number=3
43 = 64	If the last digit of perfect cube number =4, the last digit of cube root for that number=4
53 = 125	If the last digit of perfect cube number =5, the last digit of cube root for that number=5
63 = 216	If the last digit of perfect cube number =6, the last digit of cube root for that number=6
73 = 343	If the last digit of perfect cube number =3, the last digit of cube root for that number=7
83 =512	If the last digit of perfect cube number =2, the last digit of cube root for that number=8
93 = 729	If the last digit of perfect cube number =9, the last digit of cube root for that number=9
103 = 1000	If the last digit of perfect cube number =0, last digit of cube root for that number=0

Let us now see how to use the table with the help of an example. So here, we will find the Cube Root of 13824 by using the following steps.

• Step 1- Take the last three digits and make their group i.e., 13 – 824
• Step 2 Take the last digit of the above group. As per the above table, if the last digit having 4 then the last digit of cube root for that number is 4. Hence the last digit of the cube root of the given number is 4
• Step 3: Take the leftover part of the number which is 13. As per the above table, the value of 13 lies between 8 and 27 which are 2³ and 3³ respectively. 8 < 13 < 27. Take the small number i.e “ 2 “. Hence the left-most digit of the answer is 2. So our answer = 24

Solved Examples for Cube Root Formula

Example: Find Cube Root of 175616

Solution: Step 1- Take the last three digits and make their group i.e., 176 – 616

Step 2- Take the last digit of the above-formed group, i.e. 6. According to the cube rules table if the last digit having 6 then the last digit of cube root for that number is 6. Hence the last most digit of the cube root of the given number is 6

Step 3- Take the leftover group which is 176. We know that  176 lies in between the cube of the numbers i.e 5³ and 6^3. 125 < 176 < 216. Take the small cube number i.e “ 5 “. Hence the left-most digit of the answer is 5

So our answer = 56

So, using the above simple method we can calculate the cube root of a given number easily. This method can come handy while solving the questions.

Formula for Cube Root

Check the example below for detail:

5 Cube = 5³ = 125
Cube Root of 125 = ³√125 = 5

Thus, the cube root of 125 is 5. The number 125 is a perfect cube.

Cubes of Some Numbers are as Follows:

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125
• 6³ = 216
• 7³ = 343
• 8³ = 512
• 9³ = 729
• 10³ = 1000

Solved Examples Using Cube Root Formula

Question 1: What is the cube root of 1728?

Solution:

The factors of 1728 are given as,

1728 = 12 × 12 × 12

³√1728 = ³√(12 × 12 × 12)

³√1728 = 12

Properties of Cube Roots

The table given below has the cubes of all the number between 11 to 20

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From the table given above we can observe the following properties of cubes :

Property 1:

A number having 1 in its unit digit, will also have 1 in the unit digit of their cubes.

For example,

1³  =  1

11³  =  1331

21³  =  9261

31³  =  29791

Property 2:

The cubes of 1, 4, 5, 6, 9, and 0 also have the same digits in its unit digits.

For example, 

14³  =  2744

15³  =  3375

16³  =  4096

20³  =  8000

Property 3:

The cube of numbers ending 2 as unit digit will have 8 in its unit digit. Similarly, the cube of the numbers ending in unit digit 8 will have a unit digit 2.

For example, 

12³  =  1728

18³  =  5832

Property 4:

The cube of the numbers with 3 as unit digit will have a unit digit 7. Similarly, the cube of numbers with the unit digit 7 will have a unit digit 3.

For example, 

13³  =  2197

27³  =  19683

Property 5:

The cubes of all even numbers are even. Also, the cubes of odd numbers are all odd.

For example, 

18³  =  5832 (even)

27³  =  19683 (odd)

Property 6:

The sum of the cubes of the first natural numbers m is equal to the square of their sum. 

That is,

1³ + 2³ + 3³ + ……. + m³  =  (1 + 2 + 3 + ….. + m)²

For example,

1³ + 2³ + 3³ + 4³  =  (1 + 2 + 3 + 4)²

1 + 8 + 27 + 64  =  (10)²

100  =  100

Cube root of any large number can be easily found in four ways:

Let’s know how to find the cube root of any number

1. Prime factorization Method 
2. Long Division Method
3. Using Logarithms
4. Bisection Method

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