1. The problem is to simplify $\sqrt{800}$.\n\n2. Recall the property of square roots: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.\n\n3. Factor 800 into a product of a perfect square and another number: $800 = 16 \times 50$.\n\n4. Apply the square root property: $$\sqrt{800} = \sqrt{16 \times 50} = \sqrt{16} \times \sqrt{50}.$$\n\n5. Simplify $\sqrt{16}$ since 16 is a perfect square: $\sqrt{16} = 4$. So, $$\sqrt{800} = 4 \times \sqrt{50}.$$\n\n6. Next, simplify $\sqrt{50}$. Factor 50 as $25 \times 2$, where 25 is a perfect square.\n\n7. Apply the square root property again: $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \times \sqrt{2}.$$\n\n8. Substitute back: $$\sqrt{800} = 4 \times 5 \times \sqrt{2} = 20 \sqrt{2}.$$\n\n9. For an approximate decimal value, calculate $20 \times 1.4142 = 28.284$.\n\nTherefore, the simplified exact form is $20 \sqrt{2}$ and the approximate decimal value is 28.284.