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Representing Fractions

5/3 ___ 7/3. ...

Question

5/3 ___ 7/3.

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In each of the following determine rational numbers a and b :

(i) $\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = a - b \sqrt{3}$

(ii) $\frac{4 + \sqrt{2}}{2 + \sqrt{2}} = a - \sqrt{b}$

(iii) $\frac{3 + \sqrt{2}}{3 - \sqrt{2}} = a + b \sqrt{2}$

(iv) $\frac{5 + 3 \sqrt{3}}{7 + 4 \sqrt{3}} = a + b \sqrt{3}$

(v) $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} = a - b \sqrt{77}$

(vi) $\frac{4 + 3 \sqrt{5}}{4 - 3 \sqrt{5}} = a + b \sqrt{5}$

\frac{(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})} = a + \sqrt{15} b

(5 a - 7 b)^{3} + (9 c - 5 a)^{3} + (7 b - 9 c)^{3}

6 a^{3} + 5 a - 2 a^{2} - 7

9 a^{2} - 2 a - a^{3} + 2

\frac{7 + 3 \sqrt{5}}{3 + \sqrt{5}} - \frac{7 - 3 \sqrt{5}}{3 - \sqrt{5}} = a + \sqrt{5} b

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