1. The problem is to express each square root as a mixed radical, which means writing it as a product of a whole number and a square root of a smaller number. 2. The formula used is $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, where $a$ is a perfect square and $b$ is the remaining factor. 3. We find the largest perfect square factor of each number and simplify: - $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ - $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ - $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ - $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$ - $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$ - $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ - $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$ - $\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$ - $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ - $\sqrt{1000} = \sqrt{100 \times 10} = 10\sqrt{10}$ - $\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}$ - $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$ - $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$ - $\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$ - $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$ - $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ - $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$ - $\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$ - $\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}$ - $\sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3}$ 4. Each step involves identifying the largest perfect square factor and rewriting the square root accordingly. This method helps simplify square roots into mixed radicals for easier interpretation and calculation.