Properties of basic set theory The question is about a set:
$$B=\{a_1,a_2,a_3,...,a_n\} \subset \mathbb R$$
And would like to know how to calculate $B^n$ where $n \in\Bbb N$?
 A: Assuming $B = \{a_1,a_2,a_3,...,a_n\} \subset \mathbb R$, with $|B| = n$,
$B^n$ is the set of all ordered n-tuples of elements of $B$: 
The exponent refers to the operation of the Cartesian Product of $B$ with itself, n times:
$$B^n = \underbrace{B \times B \times \cdots \times B}_{\Large\text{n times}} =\{(b_1, b_2,\cdots, b_n) \mid b_i \in B\}.$$ 
Can you figure out how to calculate: $|B^n|$? There are $n$ choices for position 1, $n$ choices for position 2, ..., $n$ choices for position $n$:
$$|B^n| = \underbrace{n\times n\times \cdots \times n}_{\Large \text{n times}} = n^n$$
A: For a set $S$, we often write $S^n$ to denote the $n$-fold cartesian product of $S$ with itself. In your case, we would have
$$
B^n = \{(b_1, b_2,\ldots, b_n)\mid b_i = a_{i_k}, a_{i_k}\in B\}.
$$
In other words, $B^n$ consists of ordered $n$-tuples, where each element in the tuple is an element of the original $B$.
A: For the simpler version of what $B^n$ looks like, set up the Cartesian product of $B$'s.
$$\underbrace{B \times B \times \cdots \times B}_{\text{n times}}$$
Then, we have n-tuples consisting on elements from $B$.  As what Stahl just showed to you, we have:
$$B^n = \{(b_1, b_2, \dots, b_n) | \text{for each } b_j \in B \text{ for j} = \{1,2,3,\dots, n\} \}$$
