Cube and Cube Root
We know that the cube of a number is equal to a number multiplied by itself for three times, such as if ‘k’ is a number, then the cube of k is k \(\times\) k \(\times\) k.
The cube root of a number is another number that when multiplied by itself for three times results in the original number, such as if ‘l’ is the cube root of ‘k’, then, l \(\times\) l \(\times\) l results in ‘k’ as the product.
We use multiplication to determine the cube, and prime factorization to determine the cube root. So the process of determining the cube is the inverse of the process of determining the cube root.
Numbers that have a cube root as an integer are called perfect cube numbers.
For example, the cube root of 8 is 2, which is an integer, so 8 is a perfect cube number.
There is a direct formula to determine the cube root of perfect cube numbers. Let us explore this formula in the next section.
Formula for Cube Root
If ‘x’ is the cube of a number ‘y’, then the cube root of ‘x’ is represented by the formula as shown in the image,
Here x is a perfect cube number that can be expressed as a product of its factor y. Additionally, y is multiplied by itself thrice to give x, so y is the cube root of x.
Cubes and cube roots of a few numbers are shown in the table:
Solved Examples
Question 1: What is the cube root of 1331?
Solution:
We can write 1331 as:
1331 = 11 × 11 × 11 [Using factors of 1331]
\(\sqrt[3]{1331}=\sqrt[3]{11\times11\times11}=11\) [Cube Root Formula]
Therefore the cube root of 1331 is 11.
Example 2: Check if 729 is a perfect cube number or not.
Solution:
To figure out if 729 is a perfect cube or not, we need to express it as a product of its factors, such as,
729 = 9 x 9 x 9
\(\sqrt[3]{729}=\sqrt[3]{9\times9\times9}=9\) [Cube Root Formula]
So the cube root of 729 is 9, which is an integer.
Therefore, 729 is a perfect cube number.
Example 3: Ronald’s age is equal to the cube root of his father’s age. Determine Ronald’s age if his father is presently 27 years old.
Solution:
Present age of Ronald’s father = 27 years
Ronald’s age = cube root of 27
To find Ronald’s age, determine the cube root of 27, that is,
\(\sqrt[3]{27}=\sqrt[3]{3\times3\times3}=3\) [Cube root formula]
Therefore Ronald is currently 3 years old.
Multiplying a number thrice by itself gives us the cubed value of the number. Multiplying the number with itself twice gives the squared value of the number.
The cube root of a number can be determined using the prime factorization method and in higher grades, by applying long division.
The perfect cube numbers between 1 and 20 are 1 and 8, as the cube root of these two numbers is an integer.