1. **State the problem:** Simplify the expression $10m^2 \cdot 3m^5$ without negative exponents. 2. **Recall the rule for multiplying powers with the same base:** When multiplying terms with the same base, add the exponents: $$a^x \cdot a^y = a^{x+y}$$ 3. **Multiply the coefficients:** $$10 \cdot 3 = 30$$ 4. **Add the exponents of $m$:** $$m^2 \cdot m^5 = m^{2+5} = m^7$$ 5. **Combine the results:** $$30m^7$$ 6. **Final answer:** The simplified expression without negative exponents is $30m^7$.