Question 1 1 Simplify the following surds: \( 1.1 \quad \sqrt{75} \) \( 1.2 \quad \sqrt{50}+\sqrt{18} \) \( 1.3 \quad \frac{\sqrt{48}}{\sqrt{3}} \) \( 1.4 \quad \sqrt{27}+\sqrt{12} \) Question 2 2 Perform the following operations with surds \( 2.1 \quad \sqrt{18}+\sqrt{32} \) \( 2.2 \quad \sqrt{75}-\sqrt{12} \) \( 2.3 \quad \sqrt{6} \times \sqrt{3} \) \( 2.4 \quad \sqrt{8} \times \sqrt{2} \) \( 2.5 \quad \sqrt{18}+\sqrt{8} \) \( 2.6 \quad \sqrt{20}+\sqrt{45}-\sqrt{80} \) Question 3 3 Rationalize the denominator of the following expressions: \( 3.1 \quad \frac{5}{\sqrt{3}} \)

Calculate the value by following steps: - step0: Calculate: \(50+18\) - step1: Add the numbers: \(68\) Calculate or simplify the expression \( 18+32 \). Calculate the value by following steps: - step0: Calculate: \(18+32\) - step1: Add the numbers: \(50\) Calculate or simplify the expression \( 5/\sqrt(3) \). Calculate the value by following steps: - step0: Calculate: \(\frac{5}{\sqrt{3}}\) - step1: Multiply by the Conjugate: \(\frac{5\sqrt{3}}{\sqrt{3}\times \sqrt{3}}\) - step2: Multiply the numbers: \(\frac{5\sqrt{3}}{3}\) Calculate or simplify the expression \( 48/3 \). Calculate the value by following steps: - step0: Calculate: \(\frac{48}{3}\) - step1: Reduce the numbers: \(\frac{16}{1}\) - step2: Calculate: \(16\) Calculate or simplify the expression \( 75 \). Numbers by following steps: - step0: Solution: \(75\) Calculate or simplify the expression \( 8*2 \). Calculate the value by following steps: - step0: Calculate: \(8\times 2\) - step1: Multiply the numbers: \(16\) Calculate or simplify the expression \( 75-12 \). Calculate the value by following steps: - step0: Calculate: \(75-12\) - step1: Subtract the numbers: \(63\) Calculate or simplify the expression \( 6*3 \). Calculate the value by following steps: - step0: Calculate: \(6\times 3\) - step1: Multiply the numbers: \(18\) Calculate or simplify the expression \( 27+12 \). Calculate the value by following steps: - step0: Calculate: \(27+12\) - step1: Add the numbers: \(39\) Calculate or simplify the expression \( 18+8 \). Calculate the value by following steps: - step0: Calculate: \(18+8\) - step1: Add the numbers: \(26\) Calculate or simplify the expression \( 20+45-80 \). Calculate the value by following steps: - step0: Calculate: \(20+45-80\) - step1: Calculate: \(-15\) Let's solve the problems step by step. ### Question 1: Simplify the following surds 1.1 \( \sqrt{75} \) To simplify \( \sqrt{75} \): \[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \] 1.2 \( \sqrt{50} + \sqrt{18} \) To simplify \( \sqrt{50} \) and \( \sqrt{18} \): \[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \] \[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \] Now combine: \[ \sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2} \] 1.3 \( \frac{\sqrt{48}}{\sqrt{3}} \) To simplify \( \frac{\sqrt{48}}{\sqrt{3}} \): \[ \frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}} = \sqrt{16} = 4 \] 1.4 \( \sqrt{27} + \sqrt{12} \) To simplify \( \sqrt{27} \) and \( \sqrt{12} \): \[ \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \] \[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \] Now combine: \[ \sqrt{27} + \sqrt{12} = 3\sqrt{3} + 2\sqrt{3} = 5\sqrt{3} \] ### Question 2: Perform the following operations with surds 2.1 \( \sqrt{18} + \sqrt{32} \) To simplify: \[ \sqrt{18} = 3\sqrt{2}, \quad \sqrt{32} = 4\sqrt{2} \] Combine: \[ \sqrt{18} + \sqrt{32} = 3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2} \] 2.2 \( \sqrt{75} - \sqrt{12} \) To simplify: \[ \sqrt{75} = 5\sqrt{3}, \quad \sqrt{12} = 2\sqrt{3} \] Combine: \[ \sqrt{75} - \sqrt{12} = 5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3} \] 2.3 \( \sqrt{6} \times \sqrt{3} \) To simplify: \[ \sqrt{6} \times \sqrt{3} = \sqrt{18} = 3\sqrt{2} \] 2.4 \( \sqrt{8} \times \sqrt{2} \) To simplify: \[ \sqrt{8} \times \sqrt{2} = \sqrt{16} = 4 \] 2.5 \( \sqrt{18} + \sqrt{8} \) To simplify: \[ \sqrt{18} = 3\sqrt{2}, \quad \sqrt{8} = 2\sqrt{2} \] Combine: \[ \sqrt{18} + \sqrt{8} = 3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2} \] 2.6 \( \sqrt{20} + \sqrt{45} - \sqrt{80} \) To simplify: \[ \sqrt{20} = 2\sqrt{5}, \quad \sqrt{45} = 3\sqrt{5}, \quad \sqrt{80} = 4\sqrt{5} \] Combine: \[ \sqrt{20} + \sqrt{45} - \sqrt{80} = 2\sqrt{5} + 3\sqrt{5} - 4\sqrt{5} = 1\sqrt{5} = \sqrt{5} \] ### Question 3: Rationalize the denominator of the following expressions 3.1 \( \frac{5}{\sqrt{3}} \) To rationalize: \[ \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} \] ### Summary of Results - **Question 1:** - 1.1 \( 5\sqrt{3} \) - 1.2 \( 8\sqrt{2} \) - 1.3 \( 4 \) - 1.4 \( 5\sqrt{3} \) - **Question 2:** - 2.1 \( 7\sqrt{2} \) - 2.2 \( 3\sqrt{3} \) - 2.3 \( 3\sqrt{2} \) - 2.4 \( 4 \) - 2.5 \( 5\sqrt{2} \) - 2.6 \( \sqrt{5} \) - **Question 3:** - 3.1 \( \frac{5\sqrt{3}}{3} \)