Answer:99
The gist of this brain-teaser lies in getting your hands dirty. I first checked that for $n=1,2,3$, the product $A \times B$ is a perfect square.
Key Observation:Notice that the numbers are very close to a perfect square themselves, if $\mathbf{n} $ is even.
So, we have
$$
\begin{aligned}
15 \underbrace{99 \dots 9}_{n \ \text{times} }84 =16 \underbrace{00 \dots 0}_{(n+2) \ \text{times} } - 16 = 16 \cdot 10^{n+2 } -16. \end{aligned} $$
Similarly, we have
$$
\begin{aligned}
48 \underbrace{99\dots 9}_{n \ \text{times} } 51 = 49 \underbrace{00 \dots 0}_{(n+2) \ \text{times} } - 49= 49 \cdot 10^{n+2 } - 49
\end{aligned}
$$
So, we have
$$ A \times B = 16 (10^{n+2 }-1) \cdot 49 (10^{n+2 }-1) = (4 \cdot 7 \cdot (10^{n+2 }-1))^2. $$
Which is a perfect square for all values of $n.$ Hence, the answer is $99.$
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