1. **State the problem:** Simplify the expression $$2^{13} \cdot 15^{-3} \cdot 7^{5}$$. 2. **Recall the rules:** - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - Multiplication of powers with different bases stays as is unless bases can be factored. 3. **Rewrite the expression using the negative exponent rule:** $$2^{13} \cdot \frac{1}{15^{3}} \cdot 7^{5} = \frac{2^{13} \cdot 7^{5}}{15^{3}}$$ 4. **Factor 15:** $$15 = 3 \cdot 5$$ So, $$15^{3} = (3 \cdot 5)^{3} = 3^{3} \cdot 5^{3}$$ 5. **Rewrite the expression with factored denominator:** $$\frac{2^{13} \cdot 7^{5}}{3^{3} \cdot 5^{3}}$$ 6. **Final simplified form:** The expression cannot be simplified further since numerator and denominator have no common prime factors. **Answer:** $$\boxed{\frac{2^{13} \cdot 7^{5}}{3^{3} \cdot 5^{3}}}$$