The problem involves the Product Rule of Exponents, which states that when multiplying two expressions with the same base, you add the exponents.

Given: $7^2 \cdot 7^3$

Using the Product Rule: $7^{2 + 3} = 7^5$

So, the exponent of $7$ in its simplest exponential form is 5.

Would you like more details on how this rule works or examples?

Related questions:

1. What is the Product Rule of Exponents used for?
2. How would the rule change if the base numbers were different?
3. What happens if the exponents are negative or fractions?
4. Can this rule be applied to more than two terms, like $7^2 \cdot 7^3 \cdot 7^4$?
5. How does the Product Rule relate to the Power Rule of exponents?

Tip:

Always ensure that the base numbers are the same before applying the Product Rule of Exponents.