1. (a) Rationalize the denominator of each of the following: (i) \( \frac{2 \sqrt{3}-\sqrt{2}}{4 \sqrt{3}} \) (ii) \( \frac{x}{x+\sqrt{y}} \) (iii) \( \frac{x-\sqrt{x^{2}-9}}{x+\sqrt{x^{2}-9}} \) (v) \[ \frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)} \] (vi) \( \frac{5}{3-2 \sqrt{3}} \) (b) Rationalize the numerator in each of the following: (i) \( \frac{\sqrt{5+h}-3}{h} \) (ii) \( \frac{\sqrt{3}+\sqrt{5}}{7} \) (iii) \( \sqrt{x^{2}+1}-x \)

Simplify the expression by following steps: - step0: Solution: \(\sqrt{x^{2}+1}-x\) Calculate or simplify the expression \( (\sqrt(3)+\sqrt(5))/7 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt{3}+\sqrt{5}\right)}{7}\) - step1: Remove the parentheses: \(\frac{\sqrt{3}+\sqrt{5}}{7}\) Calculate or simplify the expression \( (\sqrt(5+h)-3)/h \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(\sqrt{5+h}-3\right)}{h}\) - step1: Remove the parentheses: \(\frac{\sqrt{5+h}-3}{h}\) Calculate or simplify the expression \( x/(x+\sqrt(y)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(x+\sqrt{y}\right)}\) - step1: Remove the parentheses: \(\frac{x}{x+\sqrt{y}}\) - step2: Simplify: \(\frac{\left(x-\sqrt{y}\right)x}{x^{2}-y}\) - step3: Multiply the terms: \(\frac{x^{2}-x\sqrt{y}}{x^{2}-y}\) Calculate or simplify the expression \( 5/(3-2*\sqrt(3)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{5}{\left(3-2\sqrt{3}\right)}\) - step1: Remove the parentheses: \(\frac{5}{3-2\sqrt{3}}\) - step2: Multiply by the Conjugate: \(\frac{5\left(3+2\sqrt{3}\right)}{\left(3-2\sqrt{3}\right)\left(3+2\sqrt{3}\right)}\) - step3: Multiply the numbers: \(\frac{5\left(3+2\sqrt{3}\right)}{-3}\) - step4: Multiply the numbers: \(\frac{15+10\sqrt{3}}{-3}\) - step5: Calculate: \(-\frac{15+10\sqrt{3}}{3}\) Calculate or simplify the expression \( (2*\sqrt(3)-\sqrt(2))/(4*\sqrt(3)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2\sqrt{3}-\sqrt{2}\right)}{\left(4\sqrt{3}\right)}\) - step1: Remove the parentheses: \(\frac{2\sqrt{3}-\sqrt{2}}{4\sqrt{3}}\) - step2: Multiply by the Conjugate: \(\frac{\left(2\sqrt{3}-\sqrt{2}\right)\sqrt{3}}{4\sqrt{3}\times \sqrt{3}}\) - step3: Multiply the numbers: \(\frac{6-\sqrt{6}}{4\sqrt{3}\times \sqrt{3}}\) - step4: Multiply the numbers: \(\frac{6-\sqrt{6}}{12}\) Calculate or simplify the expression \( 1/((\sqrt(2)+1)*(\sqrt(3)-1)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{\left(\left(\sqrt{2}+1\right)\left(\sqrt{3}-1\right)\right)}\) - step1: Remove the parentheses: \(\frac{1}{\left(\sqrt{2}+1\right)\left(\sqrt{3}-1\right)}\) - step2: Multiply the numbers: \(\frac{1}{\sqrt{6}-\sqrt{2}+\sqrt{3}-1}\) Calculate or simplify the expression \( (x-\sqrt(x^2-9))/(x+\sqrt(x^2-9)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-\sqrt{x^{2}-9}\right)}{\left(x+\sqrt{x^{2}-9}\right)}\) - step1: Remove the parentheses: \(\frac{x-\sqrt{x^{2}-9}}{x+\sqrt{x^{2}-9}}\) - step2: Simplify: \(\frac{2x^{2}-2\sqrt{x^{2}-9}\times x-9}{9}\) Let's rationalize the denominators and numerators step by step for the given expressions. ### (a) Rationalizing the Denominator #### (i) \( \frac{2 \sqrt{3}-\sqrt{2}}{4 \sqrt{3}} \) To rationalize the denominator, we can simplify the expression: \[ \frac{2 \sqrt{3}-\sqrt{2}}{4 \sqrt{3}} = \frac{6 - \sqrt{6}}{12} \] #### (ii) \( \frac{x}{x+\sqrt{y}} \) To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator: \[ \frac{x}{x+\sqrt{y}} \cdot \frac{x-\sqrt{y}}{x-\sqrt{y}} = \frac{x^{2}-x\sqrt{y}}{x^{2}-y} \] #### (iii) \( \frac{x-\sqrt{x^{2}-9}}{x+\sqrt{x^{2}-9}} \) Multiply by the conjugate: \[ \frac{x-\sqrt{x^{2}-9}}{x+\sqrt{x^{2}-9}} \cdot \frac{x-\sqrt{x^{2}-9}}{x-\sqrt{x^{2}-9}} = \frac{2x^{2}-2\sqrt{x^{2}-9}\times x-9}{9} \] #### (iv) \( \frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)} \) Multiply by the conjugate of the denominator: \[ \frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)} = \frac{1}{\sqrt{6}-\sqrt{2}+\sqrt{3}-1} \] #### (v) \( \frac{5}{3-2 \sqrt{3}} \) Multiply by the conjugate: \[ \frac{5}{3-2 \sqrt{3}} \cdot \frac{3+2\sqrt{3}}{3+2\sqrt{3}} = -\frac{15+10\sqrt{3}}{3} \] ### (b) Rationalizing the Numerator #### (i) \( \frac{\sqrt{5+h}-3}{h} \) This expression is already in a suitable form for rationalization, but we can multiply by the conjugate: \[ \frac{\sqrt{5+h}-3}{h} \cdot \frac{\sqrt{5+h}+3}{\sqrt{5+h}+3} = \frac{5+h-9}{h(\sqrt{5+h}+3)} = \frac{h-4}{h(\sqrt{5+h}+3)} \] #### (ii) \( \frac{\sqrt{3}+\sqrt{5}}{7} \) Multiply by the conjugate: \[ \frac{\sqrt{3}+\sqrt{5}}{7} \cdot \frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}} = \frac{3-5}{7(\sqrt{3}-\sqrt{5})} = \frac{-2}{7(\sqrt{3}-\sqrt{5})} \] #### (iii) \( \sqrt{x^{2}+1}-x \) Multiply by the conjugate: \[ \sqrt{x^{2}+1}-x \cdot \frac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}+x} = \frac{(x^{2}+1)-x^{2}}{\sqrt{x^{2}+1}+x} = \frac{1}{\sqrt{x^{2}+1}+x} \] ### Summary of Results 1. **Rationalized Denominators:** - (i) \( \frac{6-\sqrt{6}}{12} \) - (ii) \( \frac{x^{2}-x\sqrt{y}}{x^{2}-y} \) - (iii) \( \frac{2x^{2}-2\sqrt{x^{2}-9}\times x-9}{9} \) - (iv) \( \frac{1}{\sqrt{6}-\sqrt{2}+\sqrt{3}-1} \) - (v) \( -\frac{15+10\sqrt{3}}{3} \) 2. **Rationalized Numerators:** - (i) \( \frac{h-4}{h(\sqrt{5+h}+3)} \) - (ii) \( \frac{-2}{7(\sqrt{3}-\sqrt{5})} \) - (iii) \( \frac{1}{\sqrt{x^{2}+1}+x} \)