Question: Suppose you have the product of two expressions: $ 2^5$ \* $ 2^2$, the result will be : $ 2^{5+2}$ = $ 2^7$. This is because we know the exponent rule that if they have the same base, we can add the power. Is there a way to express the product of two expressions of different bases and powers into 1 expression with one base and power, like: $ 3^7$ \* $ 4^8$ = $ a^b$

Answer: 
No, the definition you gave is correct. Take for example $\mathbb N$ with discrete metric. Of course $\mathbb N$ is dense in it self. If you use $\mathcal B(x\_0,\varepsilon )\setminus \{x\_0\}$ instead of $\mathcal B(x\_0,\varepsilon )$, then when $\varepsilon <1$ and $n\in\mathbb N$, you'll get $$\mathcal B(n,\varepsilon )\setminus \{n\}=\emptyset,$$
what contradict density.
To rephrase the definition given,
$$
 \forall x\_0 \in B,\ \forall \varepsilon > 0,\ \exists a \in A,\ d(x\_0,a) < \varepsilon
$$
where $d$ is the metric. If $x\_0 \in A$ already, then that's fine. But what's more interesting is that points of $B \setminus A$ also have points of $A$ arbitrarily close to them.