1. **Stating the problem:** Calculate the value of the expression $$\sqrt{16} \times 3 \times \sqrt{7} \times 1 \times \sqrt{25} \times 12 \times 7 \times \sqrt{36} \times 2 \times \sqrt{1} \times 35 \times \sqrt{10^2} \times 3 \times \sqrt{225} \times 2$$. 2. **Recall the properties of square roots:** - $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ - $$\sqrt{n^2} = n$$ for any positive number $$n$$. 3. **Calculate each square root:** - $$\sqrt{16} = 4$$ - $$\sqrt{7}$$ remains as is (irrational number) - $$\sqrt{25} = 5$$ - $$\sqrt{36} = 6$$ - $$\sqrt{1} = 1$$ - $$\sqrt{10^2} = 10$$ - $$\sqrt{225} = 15$$ 4. **Substitute these values back into the expression:** $$4 \times 3 \times \sqrt{7} \times 1 \times 5 \times 12 \times 7 \times 6 \times 2 \times 1 \times 35 \times 10 \times 3 \times 15 \times 2$$ 5. **Multiply all the rational numbers (non-root terms):** Calculate stepwise: - $$4 \times 3 = 12$$ - $$12 \times 1 = 12$$ - $$12 \times 5 = 60$$ - $$60 \times 12 = 720$$ - $$720 \times 7 = 5040$$ - $$5040 \times 6 = 30240$$ - $$30240 \times 2 = 60480$$ - $$60480 \times 1 = 60480$$ - $$60480 \times 35 = 2,116,800$$ - $$2,116,800 \times 10 = 21,168,000$$ - $$21,168,000 \times 3 = 63,504,000$$ - $$63,504,000 \times 15 = 952,560,000$$ - $$952,560,000 \times 2 = 1,905,120,000$$ 6. **Now multiply by the remaining square root term $$\sqrt{7}$$:** $$1,905,120,000 \times \sqrt{7}$$ 7. **Final answer:** $$\boxed{1,905,120,000 \sqrt{7}}$$ This is the simplified exact form of the expression.