To simplify the expression using the product rule of exponents, we add the exponents together and keep the base the same.
So, we have:
\[3^8 \cdot 3^3 = 3^{8+3}.\]
Adding the exponents 8 and 3, we have:
\[3^{8+3} = 3^{11}.\]
Therefore, the expression simplifies to \boxed{3^{11}}.
Use the Product Rule Exponents to simplify the expression 3^8\:\:\cdot 3^3
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