1. **Stating the problem:** Simplify the expression $$7^5 \cdot 2^5 \times \frac{1}{14^5} \times 14^{10} \times 14^5$$. 2. **Recall the properties of exponents:** - $a^m \cdot a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(ab)^m = a^m b^m$ 3. **Rewrite $7^5 \cdot 2^5$ as $(7 \cdot 2)^5$ using the product of powers rule:** $$7^5 \cdot 2^5 = (7 \times 2)^5 = 14^5$$ 4. **Substitute back into the expression:** $$14^5 \times \frac{1}{14^5} \times 14^{10} \times 14^5$$ 5. **Simplify $14^5 \times \frac{1}{14^5}$:** $$14^5 \times \frac{1}{14^5} = \cancel{14^5} \times \frac{1}{\cancel{14^5}} = 1$$ 6. **Now the expression reduces to:** $$1 \times 14^{10} \times 14^5 = 14^{10} \times 14^5$$ 7. **Use the product of powers rule to combine:** $$14^{10} \times 14^5 = 14^{10+5} = 14^{15}$$ **Final answer:** $$14^{15}$$