The inverse operation of a cube is cube root. The symbol used to represent the cube root is \(\sqrt[3]{}\).
A cube root is a unique value that gives us the original number when we multiply itself by three times.
The cube root of \(a\) is denoted by \(\sqrt[3]{a}\) or \(a^{\frac{1}{3}}\).
Example:
Find the cube root of \(64\).
\(\sqrt[3]{64} = \sqrt[3]{4 \times 4 \times 4}\) \(= \sqrt[3]{4^3}\) \( = 4\)
Therefore, the cube root of \(64\) is \(4\).
Steps to find the cube root of a number through prime factorisation:
Step 1: Find the prime factorisation of the given number.
Step 2: Group the factors in pair of three numbers (triplet).
Step 3: If there are no factor leftover, then the given number is a perfect cube. Otherwise, it is not a perfect cube.
Step 4: Now, take one factor common from each pair and multiply them.
Step 5: The obtained product is a cube root of a given number.
1. Find the value of \(\sqrt[3]{216}\).
Solution:
Let us first find the prime factor of \(216\).
Group the factors in pair of three numbers.
\(216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)\)
Here, no factor is leftover. Therefore, \(216\) is a perfect cube.
Now, take one factor common from each pair and multiply them.
\(\sqrt[3]{216} = 2 \times 3 = 6\)
Therefore, the value of \(\sqrt[3]{216} = 6\).