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 \(Y\), 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 512 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 \(\sqrt[3]{512}\).
Step 1: Find the prime factors of the given number, that is, 512.
From above
\(512=2\times 2\times 2\times 2\times 2\times 2\times 2\times2 \times 2\)
Note: The prime factors of a number can be determined from the factor tree of the number.
Step 2: Make groups of three identical factors.
\(512=2\times 2\times 2\times 2\times 2\times 2\times 2\times2 \times 2\)
Step 3: Take one factor from each group, that is, 2, 2 and 2.
Step 4: Multiply all three factors obtained in the above step to find the cube root of the given number.
\(2\times2\ \times\ 2 = 8\)
The product obtained is the cube root of 512.
So, the cube root of 512 is 8, that is, \(\sqrt[3]{512}=8.\)
Rapid Recall
The cube root of 512 is 8.
Solved Examples
Example 1: Find the value of \(5\sqrt[3]{512}-3.\)
Solution:
\(5\sqrt[3]{512}-3\) Write the expression
\( =5\times8-3\) Substitute 8 for \(\sqrt[3]{512}\)
\( =40-3\) Multiply
\( =37\) Subtract
So, \( 5\sqrt[3]{512}-3\ =37\)
Example 2: A cuboid is divided into two identical cubes of side length a. Find the side length of each cube if the volume of the cuboid is 1024 cubic feet.
Solution:
Since the cuboid is divided into two identical cubes, the volume of each cube will be the volume of the cuboid divided by 2, that is,
volume of each cube \( =\frac{1024}{2}=512\) cubic feet.
\( V=\text{side}^3\) Write the formula for the volume of a cube
\( 512=\text{side}^3\) Substitute 512 for V
\( \sqrt[3]{512}=\text{side}\) Take cube root on both the sides
\(8=\text{side}\) Simplify
So, the side length, a, of each cube is 8 feet.
Example 3: Find the value of \( \sqrt[3]{\sqrt[3]{512}}\) .
Solution:
\( \sqrt[3]{\sqrt[3]{512}}=\sqrt[3]{8}\) substitute 8 for \( \sqrt[3]{512}\)
⇒ \( \sqrt[3]{\sqrt[3]{512}}=2\) Substitute 2 for \( \sqrt[3]{8}\)
Hence, \( \sqrt[3]{\sqrt[3]{512}} = 2.\)
[Note: The cube root of 8 is 2]
Example 4: A gift box that is cubical in shape has a volume of 512 cubic feet. Find the side length of the box.
Solution:
Use the formula for the volume of a cube to find the side length of the gift box.
\( V=\text{side}^3\) Write the formula for volume
\( 512=\text{side}^3\) Substitute 512 for V
\( \sqrt[3]{512}=\sqrt[3]{\text{side}^3}\) Take cube root on each side
\( 8=\text{side}\) Simplify
So, the side length of the gift box is 8 feet.
The cube root of -512 is -8.
\( 512^\frac{1}{3}\)
We know that the cube root of an even number is even and that of an odd number is odd. Here, 512 is an even number, so the cube root of 512 will be an even number.
Yes, because \( \sqrt[\ 3]{512}=8\) and 8 is an integer.
\(512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)