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Conditional Identities

Simplify by r...

Question

Simplify by rationalizing the denominator

(i) $\frac{4}{\sqrt{7} + \sqrt{3}}$

(ii) $\frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}}$

(iii) $\frac{5 + 3 \sqrt{2}}{5 - 3 \sqrt{2}}$

(iv) $\frac{2 \sqrt{3} - \sqrt{2}}{2 \sqrt{2} + 3 \sqrt{3}}$

(v) $\frac{7 \sqrt{3} - 5 \sqrt{2}}{\sqrt{48} + \sqrt{18}}$

(vi) $\frac{x + 9}{\sqrt{x + 9}}$

(vii) $\frac{1}{{(3 + \sqrt{5})}^{2}} + \frac{1}{{(3 - \sqrt{5})}^{2}}$

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Solution

$(i) \frac{4}{\sqrt{7} + \sqrt{3}} = \frac{4}{\sqrt{7} + \sqrt{3}} \times \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{4 (\sqrt{7} - \sqrt{3})}{{(\sqrt{7})}^{2} - {(\sqrt{3})}^{2}} = \frac{4 (\sqrt{7} - \sqrt{3})}{7 - 3} = \sqrt{7} - \sqrt{3}$

$(ii) \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} = \frac{{(\sqrt{7} - \sqrt{5})}^{2}}{{(\sqrt{7})}^{2} - {(\sqrt{5})}^{2}} = \frac{(\sqrt{7}) + (\sqrt{5}) - 2 \times \sqrt{7} \times \sqrt{5}}{7 - 5} = \frac{7 + 5 - 2 \sqrt{35}}{2} = \frac{12 - 2 \sqrt{35}}{2} = \frac{2 (6 - \sqrt{35})}{2} = 6 - \sqrt{35}$

$(iii) \frac{5 + 3 \sqrt{2}}{5 - 3 \sqrt{2}} = \frac{5 + 3 \sqrt{2}}{5 - 3 \sqrt{2}} \times \frac{5 + 3 \sqrt{2}}{5 + 3 \sqrt{2}} = \frac{{(5 + 3 \sqrt{2})}^{2}}{{(5)}^{2} - {(3 \sqrt{2})}^{2}} = \frac{5^{2} + {(3 \sqrt{2})}^{2} + 2 \times 5 \times 3 \sqrt{2}}{25 - 18} = \frac{25 + 18 + 30 \sqrt{2}}{7} = \frac{43 + 30 \sqrt{2}}{7}$

$(iv) \frac{2 \sqrt{3} - \sqrt{2}}{2 \sqrt{2} + 3 \sqrt{3}} = \frac{2 \sqrt{3} - \sqrt{2}}{2 \sqrt{2} + 3 \sqrt{3}} \times \frac{2 \sqrt{2} - 3 \sqrt{3}}{2 \sqrt{2} - 3 \sqrt{3}} = \frac{(2 \sqrt{3} - \sqrt{2}) (2 \sqrt{2} - 3 \sqrt{3})}{{(2 \sqrt{2})}^{2} - {(3 \sqrt{3})}^{2}} = \frac{2 \sqrt{3} (2 \sqrt{2} - 3 \sqrt{3}) - \sqrt{2} (2 \sqrt{2} - 3 \sqrt{3})}{8 - 27} = \frac{4 \sqrt{6} - 18 - 4 + 3 \sqrt{6}}{- 19} = \frac{22 - 7 \sqrt{6}}{19}$

$(v) \frac{7 \sqrt{3} - 5 \sqrt{2}}{\sqrt{48} + \sqrt{18}} = \frac{7 \sqrt{3} - 5 \sqrt{2}}{4 \sqrt{3} + 3 \sqrt{2}} \times \frac{4 \sqrt{3} - 3 \sqrt{2}}{4 \sqrt{3} - 3 \sqrt{2}} = \frac{(7 \sqrt{3} - 5 \sqrt{2}) (4 \sqrt{3} - 3 \sqrt{2})}{{(4 \sqrt{3})}^{2} - {(3 \sqrt{2})}^{2}} = \frac{7 \sqrt{3} (4 \sqrt{3} - 3 \sqrt{2}) - 5 \sqrt{2} (4 \sqrt{3} - 3 \sqrt{2})}{48 - 18} = \frac{84 - 21 \sqrt{6} - 20 \sqrt{6} + 30}{30} = \frac{114 - 41 \sqrt{6}}{30}$

$(vi) \frac{x + 9}{\sqrt{x + 9}} = \frac{x + 9}{\sqrt{x + 9}} \times \frac{\sqrt{x + 9}}{\sqrt{x + 9}} = \frac{(x + 9) \sqrt{x + 9}}{(x + 9)} = \sqrt{x + 9}$

$(vii) \frac{1}{{(3 + \sqrt{5})}^{2}} + \frac{1}{{(3 - \sqrt{5})}^{2}} = \frac{1}{9 + 5 + 6 \sqrt{5}} + \frac{1}{9 + 5 - 6 \sqrt{5}} = \frac{1}{14 + 6 \sqrt{5}} + \frac{1}{14 - 6 \sqrt{5}} = \frac{1}{14 + 6 \sqrt{5}} \times \frac{14 - 6 \sqrt{5}}{14 - 6 \sqrt{5}} + \frac{1}{14 - 6 \sqrt{5}} \times \frac{14 + 6 \sqrt{5}}{14 + 6 \sqrt{5}} = \frac{14 - 6 \sqrt{5}}{{(14)}^{2} - {(6 \sqrt{5})}^{2}} + \frac{14 + 6 \sqrt{5}}{{(14)}^{2} - {(6 \sqrt{5})}^{2}} = \frac{14 - 6 \sqrt{5}}{16} + \frac{14 + 6 \sqrt{6}}{16} = \frac{14 - 6 \sqrt{5} + 14 + 6 \sqrt{5}}{16} = \frac{28}{16} = \frac{7}{4}$

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