1. **State the problem:** Simplify the expression $3b^3 \cdot 8b^5$. 2. **Recall the rules:** When multiplying terms with the same base, multiply the coefficients and add the exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Multiply the coefficients:** $$3 \cdot 8 = 24$$ 4. **Add the exponents of $b$:** $$b^3 \cdot b^5 = b^{3+5} = b^8$$ 5. **Combine the results:** $$3b^3 \cdot 8b^5 = 24b^8$$ **Final answer:** $24b^8$