To simplify the expression $a^3 \cdot a^3$, you can apply the rule of exponents which states that when you multiply powers with the same base, you add the exponents:

$a^3 \cdot a^3 = a^{3+3} = a^6$

So, the simplified form is:

$a^6$

Would you like further explanation or have any other questions?

Here are five related questions that might expand your understanding:

1. How would you simplify $a^4 \cdot a^2$?
2. What is the result of $a^5 \cdot a^5$?
3. How would you approach $(a^2)^3$?
4. Can you simplify $a^3 \cdot a^2 \cdot a$?
5. What is the result of dividing $a^6$ by $a^3$?

Tip: When multiplying powers with the same base, remember to add the exponents, and when dividing, subtract the exponents.