Cube Root Formula

Cube Root Formula

Using the Cube Root Formula, one can calculate the cube root of any number. The cube root of a number is defined as the number that, when multiplied three times, becomes the original number. In radical form, a number’s cube root is represented by the symbol ∛. Using examples, let’s understand the Cube Root Formula.

Formula for Cube Root

Here is the formula for calculating the cube root: 

In the case where x is a cube root, it is equal to ∛x = ∛(y × y × y) = y

In addition, if y has an integer value, then x would be a perfect cube. 

What are Perfect Cubes?

The perfect cube of a number is the number multiplied by itself three times. Therefore, one gets a natural number when we take the Cube Root Formula of a perfect cube. 

Symbol of the Cube Root

∛ denotes the cube root, and “27 has a cube root equal to 3.”

Cubes of Some Numbers are as Follows

The following are some more examples of perfect cube numbers:

1. The perfect cube of 2 is equal to 2 × 2 × 2 = 8
2. The perfect cube of 3 equals 3 × 3 × 3 = 27
3. The perfect cube of 4 equals 4 × 4 × 4 = 64
4. The perfect cube of five equals 5 × 5 × 5 = 125 
5. The perfect cube of 6 is 6 × 6 × 6 = 216
6. The perfect cube of seven is 7 × 7 × 7 = 343
7. The perfect cube of 8 equals 8 × 8 × 8 = 512
8. The perfect cube of 9 equals 9 × 9 × 9 = 729
9. The perfect cube of 10 equals 10 × 10 × 10 = 1000

Cube Roots (For Integer Results 1 Through 10)

• One is the cube root of one
• Two is the cube root of eight
• The cube root of 27 is 3
• The cube root of 64 is 4
• 125 has a cube root of 5
• 216 has a cube root of 6
• 343 has a cube root of 7
• The cube root of 512 is 8
• 729 has a cube root of 9
• The cube root of 1000 is 10

Cube Root of Negative Numbers

Negative numbers have negative cubes as well.

−5=−5×−5×−5=−125

The cube root of -125 is -5 

Properties of Cube Roots

Property 1:

Numbers with 1 as their unit digit will also have 1 in their cubes.

The following are some examples:

1³  =  1

11³  =  1331

21³  =  9261

31³  =  29791 

Property 2:

Cubes 1, 4, 5, 6, 9, and 0 also have the same digits in their unit digits.

For example, 

14³  =  2744

15³  =  3375

16³  =  4096

20³  =  8000 

Property 3:

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

For example, 

12³  =  1728

18³  =  5832 

Property 4:

In the cube of the numbers with 3 as the unit digit, there will be a unit digit of 7. In the same way, the cube of numbers with the unit digit 7 will have a unit digit 3.

For example, 

13³  =  2197

27³  =  19683 

Property 5:

Even numbers have even cubes. Odd numbers also have odd cubes.

For example, 

18³  =  5832 (even)

27³  =  19683 (odd) 

Property 6:

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

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 

Applications of Cube Root Formula

The following are a few major applications of the Cube Root Formula:

• Equations in cubic form can be solved.
• If you are given the volume of a cube, find its dimensions.
• Give a more precise measurement of the apartment.

Cube Root Formula for Negative Numbers

Prime factorization is the best method for calculating the cube root of any number.

• Perform the prime factorization of negative numbers as well.
• The factors obtained should be divided into three groups, each containing the same number.
• The cube root can be calculated by multiplying the factors in any one group.
• It’s just that the product of three negative values gives us a negative value. In this case, the negative sign is represented by the cube root of the negative number.

To better understand the Cube Root Formula, let’s look at a couple of examples.

Examples on Cube Root Formula:

Example 1: Determine the cube root of 343

Solution:

Find the cube root of 343

Based on the Cube Root Formula,

This is the cube root of 343: 343 = (7×7×7)

= 7

343 has a cube root of 7 as its value.

Example 2: Determine whether 512 is a perfect cube.

Solution:

To find: Whether 512 is the perfect cube or not.

In order to determine the cube root, use the following formula:

Cube root of 512 = ∛512=∛(2×2×2×2×2×2×2×2×2)=∛(8×8×8)

= 8, which is an integer. Therefore, 512 is a perfect cube.

Answer: 512 is a perfect cube.

Example 3: If Ron’s age is the cube root of his grandmother’s age, and she is currently 64, calculate his age.

Solution:

To find: Ron’s age if his age is the cube root of his grandmother’s age

His grandmother’s age = 64 years.

Using the Cube Root Formula,

Ron’s age= cube root of 64 = ∛64 years = 4 years.

Answer: Ron’s age is 4 years.