EXERCISE 5 Simplify the following expressions, without the use of a calculator: \( \begin{array}{ll}\text { (1) } \sqrt{12}+\sqrt{27}-\sqrt{3} & \text { (2) } \sqrt{18}-\sqrt{50}-\sqrt{32} \\ \text { (3) } \sqrt{6} \times \sqrt{3} & \text { (4) } \frac{\sqrt{75}}{\sqrt{3}}\end{array} \)

Sure, let's simplify each expression step by step. ### **(1) \( \sqrt{12} + \sqrt{27} - \sqrt{3} \)** **Step 1:** Simplify each square root by factoring out perfect squares. - \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \) - \( \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \) - \( \sqrt{3} \) remains as is. **Step 2:** Combine like terms. - \( 2\sqrt{3} + 3\sqrt{3} - \sqrt{3} = (2 + 3 - 1)\sqrt{3} = 4\sqrt{3} \) **Final Answer:** \( 4\sqrt{3} \) --- ### **(2) \( \sqrt{18} - \sqrt{50} - \sqrt{32} \)** **Step 1:** Simplify each square root by factoring out perfect squares. - \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \) - \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \) - \( \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \) **Step 2:** Combine like terms. - \( 3\sqrt{2} - 5\sqrt{2} - 4\sqrt{2} = (3 - 5 - 4)\sqrt{2} = (-6)\sqrt{2} \) - Since radicals are typically expressed with positive coefficients, you can write it as \( -6\sqrt{2} \) or \( 6\sqrt{2} \) with a negative sign depending on the context. **Final Answer:** \( -6\sqrt{2} \) --- ### **(3) \( \sqrt{6} \times \sqrt{3} \)** **Step 1:** Use the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). - \( \sqrt{6} \times \sqrt{3} = \sqrt{6 \times 3} = \sqrt{18} \) **Step 2:** Simplify \( \sqrt{18} \). - \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \) **Final Answer:** \( 3\sqrt{2} \) --- ### **(4) \( \frac{\sqrt{75}}{\sqrt{3}} \)** **Step 1:** Simplify \( \sqrt{75} \). - \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \) **Step 2:** Divide by \( \sqrt{3} \). - \( \frac{5\sqrt{3}}{\sqrt{3}} = 5 \times \frac{\sqrt{3}}{\sqrt{3}} = 5 \times 1 = 5 \) Alternatively, you can rationalize the denominator: - \( \frac{\sqrt{75}}{\sqrt{3}} = \frac{\sqrt{75} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{225}}{3} = \frac{15}{3} = 5 \) **Final Answer:** \( 5 \) --- **Summary of Simplified Expressions:** 1. \( 4\sqrt{3} \) 2. \( -6\sqrt{2} \) 3. \( 3\sqrt{2} \) 4. \( 5 \)