Multiplication and Division of surds Wivity 8 Simplify the following expressions: \( \begin{array}{ll}\text { a) } 2 \sqrt{3} \cdot 5 \sqrt{3} & \text { b) }(3 \sqrt{2}+3)(3 \sqrt{2}-3) \\ \text { c) } \frac{\sqrt{50}+\sqrt{18}}{\sqrt{32}} & \text { d) } \sqrt{21} \cdot \sqrt{60} \cdot \sqrt{35}\end{array} \) Determine, without the use of the calculator, which is greater \( \sqrt{7} \) or \( \sqrt[3]{15} \) ?

Let's simplify each of the given expressions step by step and then compare \( \sqrt{7} \) and \( \sqrt[3]{15} \). --- ### a) \( 2 \sqrt{3} \cdot 5 \sqrt{3} \) **Solution:** Multiply the coefficients and the radicals separately. \[ 2 \sqrt{3} \cdot 5 \sqrt{3} = (2 \times 5) \times (\sqrt{3} \times \sqrt{3}) = 10 \times \sqrt{9} = 10 \times 3 = 30 \] **Answer:** \( 30 \) --- ### b) \( (3 \sqrt{2} + 3)(3 \sqrt{2} - 3) \) **Solution:** Recognize the expression as a difference of squares: \( (a + b)(a - b) = a^2 - b^2 \), where \( a = 3\sqrt{2} \) and \( b = 3 \). \[ (3 \sqrt{2})^2 - (3)^2 = 9 \times 2 - 9 = 18 - 9 = 9 \] **Answer:** \( 9 \) --- ### c) \( \frac{\sqrt{50} + \sqrt{18}}{\sqrt{32}} \) **Solution:** Simplify each square root: \[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \\ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \\ \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \] Substitute back into the expression: \[ \frac{5\sqrt{2} + 3\sqrt{2}}{4\sqrt{2}} = \frac{8\sqrt{2}}{4\sqrt{2}} = \frac{8}{4} = 2 \] **Answer:** \( 2 \) --- ### d) \( \sqrt{21} \cdot \sqrt{60} \cdot \sqrt{35} \) **Solution:** Combine the radicals: \[ \sqrt{21} \cdot \sqrt{60} \cdot \sqrt{35} = \sqrt{21 \times 60 \times 35} \] Calculate the product inside the radical: \[ 21 \times 60 = 1260 \\ 1260 \times 35 = 44100 \\ \sqrt{44100} = \sqrt{441 \times 100} = \sqrt{441} \times \sqrt{100} = 21 \times 10 = 210 \] **Answer:** \( 210 \) --- ### Comparing \( \sqrt{7} \) and \( \sqrt[3]{15} \) **Objective:** Determine which is greater without using a calculator. **Approach:** Calculate approximate decimal values. 1. **Calculate \( \sqrt{7} \):** \[ \sqrt{7} \approx 2.6458 \] 2. **Calculate \( \sqrt[3]{15} \):** We know that \( 2.466^3 \approx 15 \) (since \( 2.466^3 \approx 2.466 \times 2.466 \times 2.466 \approx 15 \)). \[ \sqrt[3]{15} \approx 2.466 \] **Comparison:** \[ \sqrt{7} \approx 2.6458 > \sqrt[3]{15} \approx 2.466 \] **Conclusion:** \( \sqrt{7} \) is greater than \( \sqrt[3]{15} \). --- **Final Summary:** - **a)** \( 30 \) - **b)** \( 9 \) - **c)** \( 2 \) - **d)** \( 210 \) - **Comparison:** \( \sqrt{7} > \sqrt[3]{15} \)