EXERCISE 5 (a) Simplify the following expressions, without the use of a calculator: (1) \( \sqrt{12}+\sqrt{27}-\sqrt{3} \) (2) \( \sqrt{18}-\sqrt{50}-\sqrt{32} \) (3) \( \sqrt{6} \times \sqrt{3} \) (4) \( \frac{\sqrt{75}}{\sqrt{3}} \) (5) \( (\sqrt{3}+\sqrt{12})^{2} \) (6) \( (\sqrt{3}+\sqrt{5})(\sqrt{3}-\sqrt{5}) \) (7) \( \sqrt{3}\left(\sqrt{12}-\sqrt{1 \frac{1}{3}}\right) \) (8) \( \frac{\sqrt{50}+\sqrt{2}}{\sqrt{18}} \) (9) \( \frac{6 \sqrt{8}+4 \sqrt{18}}{3 \sqrt{32}-\sqrt{72}} \) (10) \( \frac{2 \sqrt{12}+4 \sqrt{20}}{\sqrt{12}+\sqrt{80}} \) 38

Calculate the value by following steps: - step0: Calculate: \(\frac{\sqrt{75}}{\sqrt{3}}\) - step1: Simplify the root: \(\frac{5\sqrt{3}}{\sqrt{3}}\) - step2: Reduce the fraction: \(5\) Calculate or simplify the expression \( \sqrt(6)*\sqrt(3) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{6}\times \sqrt{3}\) - step1: Use the properties of radicals: \(\sqrt{6\times 3}\) - step2: Calculate: \(\sqrt{18}\) - step3: Write as a product: \(\sqrt{9\times 2}\) - step4: Write in exponential form: \(\sqrt{3^{2}\times 2}\) - step5: Use the properties of radicals: \(\sqrt{3^{2}}\times \sqrt{2}\) - step6: Simplify the root: \(3\sqrt{2}\) Calculate or simplify the expression \( (\sqrt(3)+\sqrt(5))(\sqrt(3)-\sqrt(5)) \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)\) - step1: Simplify the product: \(\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}\) - step2: Evaluate the power: \(3-5\) - step3: Subtract the numbers: \(-2\) Calculate or simplify the expression \( (\sqrt(3)+\sqrt(12))^2 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt{3}+\sqrt{12}\right)^{2}\) - step1: Simplify the root: \(\left(\sqrt{3}+2\sqrt{3}\right)^{2}\) - step2: Add the numbers: \(\left(3\sqrt{3}\right)^{2}\) - step3: Evaluate the power: \(9\times 3\) - step4: Multiply the numbers: \(27\) Calculate or simplify the expression \( \sqrt(12)+\sqrt(27)-\sqrt(3) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{12}+\sqrt{27}-\sqrt{3}\) - step1: Simplify the root: \(2\sqrt{3}+\sqrt{27}-\sqrt{3}\) - step2: Simplify the root: \(2\sqrt{3}+3\sqrt{3}-\sqrt{3}\) - step3: Collect like terms: \(\left(2+3-1\right)\sqrt{3}\) - step4: Calculate: \(4\sqrt{3}\) Calculate or simplify the expression \( (\sqrt(50)+\sqrt(2))/\sqrt(18) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt{50}+\sqrt{2}\right)}{\sqrt{18}}\) - step1: Remove the parentheses: \(\frac{\sqrt{50}+\sqrt{2}}{\sqrt{18}}\) - step2: Simplify the root: \(\frac{5\sqrt{2}+\sqrt{2}}{\sqrt{18}}\) - step3: Simplify the root: \(\frac{5\sqrt{2}+\sqrt{2}}{3\sqrt{2}}\) - step4: Add the numbers: \(\frac{6\sqrt{2}}{3\sqrt{2}}\) - step5: Reduce the fraction: \(2\) Calculate or simplify the expression \( \sqrt(18)-\sqrt(50)-\sqrt(32) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{18}-\sqrt{50}-\sqrt{32}\) - step1: Simplify the root: \(3\sqrt{2}-\sqrt{50}-\sqrt{32}\) - step2: Simplify the root: \(3\sqrt{2}-5\sqrt{2}-\sqrt{32}\) - step3: Simplify the root: \(3\sqrt{2}-5\sqrt{2}-4\sqrt{2}\) - step4: Collect like terms: \(\left(3-5-4\right)\sqrt{2}\) - step5: Subtract the numbers: \(-6\sqrt{2}\) Calculate or simplify the expression \( \sqrt(3)*(\sqrt(12)-\sqrt(1+1/3)) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{3}\times \left(\sqrt{12}-\sqrt{1+\frac{1}{3}}\right)\) - step1: Simplify the root: \(\sqrt{3}\times \left(2\sqrt{3}-\sqrt{1+\frac{1}{3}}\right)\) - step2: Add the numbers: \(\sqrt{3}\times \left(2\sqrt{3}-\sqrt{\frac{4}{3}}\right)\) - step3: Simplify the root: \(\sqrt{3}\times \left(2\sqrt{3}-\frac{2\sqrt{3}}{3}\right)\) - step4: Subtract the numbers: \(\sqrt{3}\times \frac{4\sqrt{3}}{3}\) - step5: Multiply: \(\frac{\sqrt{3}\times 4\sqrt{3}}{3}\) - step6: Multiply: \(\frac{12}{3}\) - step7: Reduce the fraction: \(4\) Calculate or simplify the expression \( (2*\sqrt(12)+4*\sqrt(20))/(\sqrt(12)+\sqrt(80)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2\sqrt{12}+4\sqrt{20}\right)}{\left(\sqrt{12}+\sqrt{80}\right)}\) - step1: Remove the parentheses: \(\frac{2\sqrt{12}+4\sqrt{20}}{\sqrt{12}+\sqrt{80}}\) - step2: Simplify the root: \(\frac{2\times 2\sqrt{3}+4\sqrt{20}}{\sqrt{12}+\sqrt{80}}\) - step3: Simplify the root: \(\frac{2\times 2\sqrt{3}+4\times 2\sqrt{5}}{\sqrt{12}+\sqrt{80}}\) - step4: Multiply the terms: \(\frac{4\sqrt{3}+4\times 2\sqrt{5}}{\sqrt{12}+\sqrt{80}}\) - step5: Multiply the terms: \(\frac{4\sqrt{3}+8\sqrt{5}}{\sqrt{12}+\sqrt{80}}\) - step6: Simplify the root: \(\frac{4\sqrt{3}+8\sqrt{5}}{2\sqrt{3}+\sqrt{80}}\) - step7: Simplify the root: \(\frac{4\sqrt{3}+8\sqrt{5}}{2\sqrt{3}+4\sqrt{5}}\) - step8: Rewrite the expression: \(\frac{\left(2\sqrt{3}+4\sqrt{5}\right)\times 2}{2\sqrt{3}+4\sqrt{5}}\) - step9: Reduce the fraction: \(2\) Calculate or simplify the expression \( (6*\sqrt(8)+4*\sqrt(18))/(3*\sqrt(32)-\sqrt(72)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6\sqrt{8}+4\sqrt{18}\right)}{\left(3\sqrt{32}-\sqrt{72}\right)}\) - step1: Remove the parentheses: \(\frac{6\sqrt{8}+4\sqrt{18}}{3\sqrt{32}-\sqrt{72}}\) - step2: Simplify the root: \(\frac{6\times 2\sqrt{2}+4\sqrt{18}}{3\sqrt{32}-\sqrt{72}}\) - step3: Simplify the root: \(\frac{6\times 2\sqrt{2}+4\times 3\sqrt{2}}{3\sqrt{32}-\sqrt{72}}\) - step4: Simplify the root: \(\frac{6\times 2\sqrt{2}+4\times 3\sqrt{2}}{3\times 4\sqrt{2}-\sqrt{72}}\) - step5: Multiply the terms: \(\frac{12\sqrt{2}+4\times 3\sqrt{2}}{3\times 4\sqrt{2}-\sqrt{72}}\) - step6: Multiply the terms: \(\frac{12\sqrt{2}+12\sqrt{2}}{3\times 4\sqrt{2}-\sqrt{72}}\) - step7: Simplify the root: \(\frac{12\sqrt{2}+12\sqrt{2}}{3\times 4\sqrt{2}-6\sqrt{2}}\) - step8: Multiply the terms: \(\frac{12\sqrt{2}+12\sqrt{2}}{12\sqrt{2}-6\sqrt{2}}\) - step9: Add the numbers: \(\frac{24\sqrt{2}}{12\sqrt{2}-6\sqrt{2}}\) - step10: Subtract the numbers: \(\frac{24\sqrt{2}}{6\sqrt{2}}\) - step11: Reduce the fraction: \(4\) Here are the simplified results for each expression: 1. **Expression:** \( \sqrt{12}+\sqrt{27}-\sqrt{3} \) **Simplified Result:** \( 4\sqrt{3} \) 2. **Expression:** \( \sqrt{18}-\sqrt{50}-\sqrt{32} \) **Simplified Result:** \( -6\sqrt{2} \) 3. **Expression:** \( \sqrt{6} \times \sqrt{3} \) **Simplified Result:** \( 3\sqrt{2} \) 4. **Expression:** \( \frac{\sqrt{75}}{\sqrt{3}} \) **Simplified Result:** \( 5 \) 5. **Expression:** \( (\sqrt{3}+\sqrt{12})^{2} \) **Simplified Result:** \( 27 \) 6. **Expression:** \( (\sqrt{3}+\sqrt{5})(\sqrt{3}-\sqrt{5}) \) **Simplified Result:** \( -2 \) 7. **Expression:** \( \sqrt{3}\left(\sqrt{12}-\sqrt{1 \frac{1}{3}}\right) \) **Simplified Result:** \( 4 \) 8. **Expression:** \( \frac{\sqrt{50}+\sqrt{2}}{\sqrt{18}} \) **Simplified Result:** \( 2 \) 9. **Expression:** \( \frac{6 \sqrt{8}+4 \sqrt{18}}{3 \sqrt{32}-\sqrt{72}} \) **Simplified Result:** \( 4 \) 10. **Expression:** \( \frac{2 \sqrt{12}+4 \sqrt{20}}{\sqrt{12}+\sqrt{80}} \) **Simplified Result:** \( 2 \) If you need further explanations or details on any specific step, feel free to ask!