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