Cube Root Formula

To know the cube root formula, you need to know the cube formula.

Cube of any digit, forms by multiplying the digit by itself three times. For instance, to find the cube:

We need to multiply 5 three times in the case of 53: 5 × 5 × 5 = 125

Note: We need to write down “5 cube” as 53(the little 3 on the top means the number appears three times during the multiplication process.)

The Cube Formula for any value ‘x’ can be given as,

\[x^{3} = x \times x \times x\]


What are Perfect Cubes?

If we group the prime factors of a number in triples of equal factors, then that number is known as a perfect cube. In order to check whether a number is a perfect cube or not, we need to find its prime factors and then group together triplets of the prime factors. If no factor is left out after the process then the number is known to be a perfect cube.

Here are a few more examples of perfect cube numbers:

  1. Perfect cube of 2 equals 2 × 2 × 2 = 8

  2. Perfect cube of 3 equals 3 × 3 × 3 = 27

  3. Perfect cube of 4 equals 4 × 4 × 4 = 64

  4. Perfect cube of 5 equals 5 × 5 × 5 = 125 

  5. Perfect cube of 6 equals 6 × 6 × 6 = 216

  6. Perfect cube of 7 equals 7 × 7 × 7 = 343

  7. Perfect cube of 8 equals 8 × 8 × 8 = 512

  8. Perfect cube of 9 equals 9 × 9 × 9 = 729

  9. Perfect cube of 10 equals 10 × 10 × 10 = 1000

What is a Cube Root?

A cube root goes the opposite direction. For example, 3 is cubed to give the result 27 so the cube root of 27 will be 3.

Therefore, the cube root of a number is a special number which when cubed gives the original number as a result. The cube root of 27 is 3 because 3 is cubed to produce 27.


Symbol of the Cube Root

∛ is the symbol used to denote the cube root. We can use it as \[\sqrt[3]{27} = 3\] and we read it as “the cube root of 27 equals 3”.


The Formula for Cube Root

Check the example below for detail:

5 Cube = 53= 125

Cube Root of 125 = \[\sqrt[3]{125} = 5\]

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


Cube Roots (For Integer Results 1 Through 10)

  • Cube root of 1 is 1

  • Cube root of 8 is 2

  • Cube root of 27 is 3

  • Cube root of 64 is 4

  • Cube root of 125 is 5

  • Cube root of 216 is 6

  • Cube root of 343 is 7

  • Cube root of 512 is 8

  • Cube root of 729 is 9

  • Cube root of 1000 is 10

Cube Root of Negative Numbers

The cube of a negative number will also be a negative number.

\[- 5 = - 5 \times -5 \times -5 = - 125\]

\[\sqrt[3]{-125} = -5\]


Properties of Cube Roots

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


(image will be uploaded soon)


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

Solved Questions

Question 1)What is the cube root of 1728?

Solution:

The factors of 1728 are given as,

\[1728 = 12 \times 12 \times 12\]

\[\sqrt[3]{1728} = \sqrt[3]{12 \times 12 \times 12}\]

\[\sqrt[3]{1728} = 12\]


Question 2) Find the cube root of the following:

  • 27

  • 125

Solution: Cube root of 27 is 3. (3 × 3 × 3)

Cube root of 125 is 5 (5 × 5 × 5)

FAQ (Frequently Asked Questions)

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.