What is Cube Root?
We already know that the cube root of a number is a value that when multiplied by itself three times results in the original number. For example, when a number x is multiplied by itself three times we get a number y, so we can say that y is a cube of x or x is the cube root of y.
When writing the cube root, we write it as a symbol, ‘\(\sqrt[3]{~}\)’
Mathematically, \(\sqrt[3]{y}\) = x (cube root of y is x)
In this article we have the number 216 whose cube root is 6, so we can write this as a relation,
\(\sqrt{216}=6\)
Now let us see how we can obtain the cube root of 216.
How do we find the Cube Root of 216?
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 \(\sqrt[3]{216}.\)
Step 1: Find the prime factors of the given number, that is, 216.
From the factor tree we can observe that,
Prime factorization of 216 = 2 x 2 x 2 x 3 x 3 x 3
Note: Prime factors of a number can always be determined from the factor tree of the number.
Step 2: Group three identical factors in one group.
216 = 2 x 2 x 2 x 3 x 3 x 3
Step 3: Take one factor from each group, which is, 2 and 3.
Step 4: Multiply the two factors obtained in the above step to find the cube root of the given number.
\(\sqrt[3]{216}=2\times3\)
\(\sqrt[3]{216}=6\)
So, the cube root of 216 is 6.
Rapid Recall
The cube root of 216 is 6.
Solved Examples
Example 1: The volume of a spherical iron ball is 904.32 cubic inches. Find the radius of the ball. Use 3.14 for \(\pi.\)
Solution:
V = \(\frac{4}{3}\pi r^3\) Write the formula for volume of sphere
\(904.32 = \frac{4}{3}\times3.14\times r^3\) Substitute 904.32 for V and 3.14 for \(\pi.\)
\(904.32\times3 = 4\times3.14\times r^3\) Multiply both side by 3
\(\frac{904.32~\times~3}{4~\times~3.14}=\frac{4~\times 3.14~\times ~r^3}{4~\times~3.14}\) Divide each side by (4 x 3.14)
\(216 = r^3\) Simplify
\(\sqrt[3]{216}=\sqrt[3]{r^3}\) Take cube root on both sides
\(6 = r \) Simplify
So, the radius of the spherical ball is 6 inches.
Example 2: Find the value of \(3\sqrt[3]{216}-\sqrt[3]{125}.\)
Solution:
\(3\sqrt[3]{216}-\sqrt[3]{125}\)
⇒ \(3\times6-5\)
⇒ \(18-5\)
⇒ 13
So, \(3\sqrt[3]{216}-\sqrt[3]{125}=13.\)
Example 3: Find the cube root of 125.
Solution: Write down the prime factors of 125 using the factor tree method.
From above
Prime factorization of 125 = 5 x 5 x 5
\(\sqrt[3]{125}=5\)
So, the cube root of 125 is 5.
Example 4: A gift box that is cubical in shape has a volume of 216 cubic inches. Find the edge length of the box.
Solution:
Use the volume of the cube formula to find the edge length of the cube.
V = \(\text{side}^3\) Write the formula for volume
216 = \(\text{side}^3\) Substitute 216 for V
\(\sqrt[3]{216}=\sqrt[3]{\text{side}^3}\) Take cube root on each side
\(6 =\text{side}\) Simplify
So, the edge length of the gift box is 6 inches.
In the exponential form, the cube root of 216 is written as 216 raised to the power of \(\frac{1}{3}\).
Since the cube root of 216 is 6, therefore, two times the cube root, is 12.
Yes, 216 is a perfect cube and when we apply prime factorization, we get 6 as the cube root, which is a whole number and an integer.
As we know that the cube root of an even number is always even. 216 is an even number, so the cube root of 216 will be an even number.
It is known that two negative numbers when multiplied result in a positive number. However, when a negative number is cubed, it results in a negative cube, and conversely, the cube root of a negative number, is also negative.