Cube and Cube Root
As already explained, the process of finding the cube root of a number is the opposite of finding the cube.
In general, if a number ‘X’ is multiplied three times by itself to get ‘Y’, then the cube of X is denoted by X\(^3\), which is equal to Y. On the other hand, the cube root of Y is denoted by \(\sqrt[3]{y}\) which is equal to X.
Mathematically, we always represent cubes and cube roots as, X\(^3\) = Y and \(\sqrt[3]{y}\) = X.
Note : The cube root of a number is denoted by ‘\(\sqrt[3]{~~~ .}\)’
Cube root of 64 using Prime Factorization
The following steps can be applied to determine the cube root of any number using the prime factorization method. In this case, this method is applied to find \(\sqrt[3]{64}\).
Step 1: Find the prime factors of the given number, that is, 64.
So, the prime factorization of 64 is:
64 = 2 x 2 x 2 x 2 x 2 x 2
Note: Prime factors of a number can be determined from the factor tree of the number.
Step 2: Group three identical factors in one group.
64 = 2 x 2 x 2 x 2 x 2 x 2
Step 3: Take one factor from each group, that is, 2 and 2.
Step 4: Multiply the two factors obtained in the above step to find the cube root of the given number.
2 x 2 = 4
Hence, the cube root of 64 is 4, that is, \(\sqrt[3]{64}\) = 4.
Rapid Recall
The cube root of 64 is 4.
Solved Examples
Example 1: Find the value of 5 \(\sqrt[3]{64}\) – 2 \(\sqrt[3]{27}\).
Solution:
5 \(\sqrt[3]{64}\) – 2 \(\sqrt[3]{27}\) [Write the expression]
⇒ 5 x 4 – 2 x 3 [Substitute 4 for \(\sqrt[3]{64}\) and 3 for \(\sqrt[3]{27}\)]
⇒ 20 – 6 [Multiply]
⇒ 14 [Subtract]
So, 5 \(\sqrt[3]{64}\) – 2 \(\sqrt[3]{27}\) = 14.
Example 2: A cubical box has a volume of 64 cubic inches. Find the edge length of the box.
Solution:
V = side\(^3\) [Write the formula for volume of cube]
64 = side\(^3\) [Substitute 64 for V]
\(\sqrt[3]{64}\) = \(\sqrt[3]{side}^3\) [Apply cube root on both sides]
4 = side [Find cube root of 64]
Therefore, the edge length of the cubical box is 4 inches.
Example 3: Find the value of y if \(\frac{1}{2}y^3\) = – 32.
Solution:
\(\frac{1}{2}y^3\) = – 32 [Write the equation]
2 x \(\frac{1}{2}y^3\) = – 32 x 2 [Multiply each side by 2]
\(y^3\) = – 64 [solve]
\(\sqrt[3]{y^3}\) = \(\sqrt[3]{-~64}\) [Take the cube of each side]
y = – 4 [Simplify]
Hence, the value of y is -4.
The cube root of -64 is -4, and -4 is an integer.
Hence, -64 is a perfect cube.
Cube of 1 = 1
Cube of 2 = 8
Cube of 3 = 27
Cube of 4 = 64
Cube of 5 = 125
From the above, four perfect cubes(1, 8, 27 and 64) lie between 0 and 80.
Since, the cube root of 64 = 4
So, thrice the cube root of 64 is 12.
No, the cube root of a positive integer is always a positive number. Conversely, the cube root of a negative integer is always a negative number.