1. **State the problem:** We want to rewrite radicals as products of radicals and simplify them. 2. **Recall the property of square roots:** For positive real numbers $a$ and $b$, $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$ This means we can break down a square root of a product into the product of square roots. 3. **Rewrite each radical as a product of radicals in two different ways and check decimal equivalence:** - For $\sqrt{18}$: - Decimal: $\sqrt{18} \approx 4.2426$ - Product #1: $\sqrt{9} \times \sqrt{2} = 3 \times \sqrt{2}$ - Product #2: $\sqrt{6} \times \sqrt{3}$ - Simplified radical: $3\sqrt{2}$ - For $\sqrt{44}$: - Decimal: $\sqrt{44} \approx 6.6332$ - Product #1: $\sqrt{4} \times \sqrt{11} = 2 \times \sqrt{11}$ - Product #2: $\sqrt{22} \times \sqrt{2}$ - Simplified radical: $2\sqrt{11}$ - For $\sqrt{20}$: - Decimal: $\sqrt{20} \approx 4.4721$ - Product #1: $\sqrt{4} \times \sqrt{5} = 2 \times \sqrt{5}$ - Product #2: $\sqrt{10} \times \sqrt{2}$ - Simplified radical: $2\sqrt{5}$ - For $\sqrt{45}$: - Decimal: $\sqrt{45} \approx 6.7082$ - Product #1: $\sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5}$ - Product #2: $\sqrt{15} \times \sqrt{3}$ - Simplified radical: $3\sqrt{5}$ 4. **Simplify radicals to whole numbers:** - Radicals that simplify to whole numbers have radicands that are perfect squares (like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100). - When the radicand contains a perfect square factor, we can take its square root out of the radical. 5. **How the product leads to the simplified radical:** - For example, $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$. - We separate the radicand into a perfect square and another factor, then simplify the perfect square. 6. **How Jordan simplified $\sqrt{40}$ into $2\sqrt{10}$:** - $\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$. 7. **Check the given examples c. and d.:** - c. $\sqrt{24} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$ - d. $\sqrt{24} = \sqrt{24} \times \sqrt{1} = \sqrt{24}$ (no simplification here) This confirms the property and simplification process.