Study the pattern in these examples.
\[
\begin{array}{l}
a^{2} \# a^{5}=a^{12} \\
a^{6} \# a^{4}=a^{26} \\
a^{5} \# a^{7}=a^{37}
\end{array}
\]
Select the equation that describes the pattern.
\[
\begin{array}{l}
a^{x} \# a^{y}=a^{2 x+y} \\
a^{x} \# a^{y}=a^{x+2 y} \\
a^{x} \# a^{y}=a^{x+y+4} \\
a^{x} \# a^{y}=a^{x \cdot y+2}
\end{array}
\]
Select the correct equation.
A. $a^{x} \# a^{y}=a^{x \cdot y+2}$
B. $a^{x} \# a^{y}=a^{x+y+4}$
C. $a^{x}=a^{y}=a^{x+2 y}$.
D. $a^{x}=a^{y}=a^{2 x+y}$

Step 1 ：Observe the given equations: \(a^{2} \# a^{5}=a^{12}\), \(a^{6} \# a^{4}=a^{26}\), and \(a^{5} \# a^{7}=a^{37}\).

Step 2 ：Notice that the exponent on the right side of the equation is the product of the exponents on the left side plus 2.

Step 3 ：For example, in the first equation, \(a^{2} \# a^{5}=a^{12}\), the exponent on the right side (12) is the product of the exponents on the left side (2 and 5) plus 2.

Step 4 ：This pattern seems to hold for the other equations as well.

Step 5 ：Therefore, the equation that describes the pattern is \(a^{x} \# a^{y}=a^{x \cdot y+2}\).

Step 6 ：Final Answer: The correct equation that describes the pattern is \(\boxed{a^{x} \# a^{y}=a^{x \cdot y+2}}\).