Find the set-builder form [rule method] of \[{B_4} = \left\{ {8,27,64,125,216} \right\}\].

Hint: We can represent a set in the set-builder form if all the elements of that set possess a common property that is not possessed by any element outside the set. We need to identify that common property and represent the given set accordingly.

Complete step-by-step answer:
We will study the elements of the given set carefully and try to find a common property between them.
The first element of the set is 8. 8 can also be represented as \[{2^3}\].
The second element of the set is 27. 27 can also be represented as \[{3^3}\].
The third element of the set is 64. 64 can also be represented as \[{4^3}\].
Similarly, the fourth element of the set 125 can be represented as \[{5^3}\] and the fifth element of the set 216 can be represented as \[{6^3}\].
We can observe that all elements of the set are cubes of natural numbers starting from 2 till 6.
Now, we will represent this information in the set-builder form.
We will describe the elements by the symbol \[x\] which will be followed by a colon [:]. After the colon sign, we will write the common property of all the elements of the set and enclose everything in curly brackets. The colon symbol is read as ‘such that’ and the curly brackets stand for ‘set of all’.
Let us write the set-builder form of the given set:
\[{B_4} = \{ x:x = {n^3}\], where \[n\] is a natural number and \[2 \le n \le 6\} \].
We will read the above representation as the set of all \[x\] such that \[x\] is the cube of a natural number lying between 2 and 6.

Note: The set can be represented both in the form of a statement or in form of symbols or both. For example, the given set can also be represented in the following manner:
\[{B_4} = \{ x:x = {n^3}\] where \[n \in N\] and \[2 \le n \le 6\} \]
Here \[N\] represents the set of natural numbers.
The set-builder form is also known as the rule method because a rule or condition is fixed and all elements of the set must adhere to that rule.