Cube and Cube Root
As described earlier, the cube root of a number is a number that when multiplied by itself three times results in the original number. Hence, the process for finding the cube root of a number is the opposite of finding the cube, the main difference being whether we multiply or divide!
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 729 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 \(\sqrt[3]{729} \).
Step 1: Find the prime factors of the given number, that is, 729.
So the prime factorization of 729 is:
\(729~=~3~\times~3~\times~3~\times~3~\times~3~\times~3 \)
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.
Step 3: Take one factor from each group, that is, 3 and 3.
Step 4: Multiply the two factors obtained in the above step to find the cube root of the given number.
\(3~\times~3~=~9 \)
The product obtained is the cube root of 729.
So, the cube root of 729 is 9, that is, \(\sqrt[3]{729}~=~9 \).
Rapid Recall
The cube root of 729 is 9.
Solved Examples
Example 1: Two identical rooms cubical in shape have a total volume of 1458 cubic feet. Find the dimensions of these rooms.
Solution: Volume of two rooms = 1458 cubic feet
So, the volume of each room \(~=~\frac{1458}{2}~=~729 \) cubic feet.
Let us assume the dimensions of each room is n feet x n feet x n feet.
Volume of room \(~=~n^3~=~729\) [Volume of cube formula]
\(n~=~\sqrt[3]{729}\) [Cube root on both sides]
\(\sqrt[3]{729}~=~9\)
So \(n~=~9\)
So, each of the rooms has the dimensions of 9 feet \(\times\) 9 feet \(\times\) 9 feet.
Example 2: A gift box that is cubical in shape has a volume of 729 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
\(729~=~{\text{side}}^3\) Substitute 729 for V
\(\sqrt[3]{729}~=~\sqrt[3]{{\text{side}}^3}\) Take cube root on each side
\(9~=~\text{side}\) Simplify
So, the edge length of the gift box is 9 inches.
Example 3: Find the value of 4\(\sqrt[3]{729}~-~25\).
Solution:
4\(\sqrt[3]{729}~-~25\) Write the expression
⇒ \(4~\times~9~-~25\) Substitute 9 for \(\sqrt[3]{729}\)
⇒ \(36~-~25\) Multiply
⇒ 11 Subtract
So, 4\(\sqrt[3]{729}~-25~=~11\)
(-9) \(\times\) (-9) \(\times\) (-9) = -729
So, the cube root of -729 is -9.
Cube root of x is denoted of \(\sqrt[3]{x}\) or \(x^{\frac{1}{3}}\).
\((-729)^{\frac{1}{3}}\).
\(1^3~=~1\)
\(2^3~=~8\)
\(3^3~=~27\)
\(4^3~=~64\)
\(5^3~=~125\)
From above only 64 lies between 50 and 100 which is a perfect cube.
So, only one perfect cube lies between 50 and 100.