To simplify the expression \( 7^2 \cdot 7^3 \) using the Product Rule of Exponents, we apply the rule which states that when we multiply two exponential expressions with the same base, we add the exponents.
Thus, we have:
\[ 7^2 \cdot 7^3 = 7^{2 + 3} \]
Calculating the exponent:
\[ 2 + 3 = 5 \]
Therefore, the expression simplifies to:
\[ 7^5 \]
So, the exponent of \( 7^2 \cdot 7^3 \) in exponential simplest form is \( \boxed{5} \).