Express y2/root of x2+y2 + x with rational denominator 2. Simplify (3/5) 4 (8/5) 12 (32/5) 6 3.if a=3+ root7/2 find the value of a2+1/a2

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Solution

$(1) \frac{y^{2}}{\sqrt{x^{2} + y^{2}} + x} = \frac{y^{2}}{\sqrt{x^{2} + y^{2}} + x} \times \frac{\sqrt{x^{2} + y^{2}} - x}{\sqrt{x^{2} + y^{2}} - x} = \frac{y^{2} (\sqrt{x^{2} + y^{2}} - x)}{{(\sqrt{x^{2} + y^{2}})}^{2} - {(x)}^{2}} = \frac{y^{2} (\sqrt{x^{2} + y^{2}} - x)}{x^{2} + y^{2} - x^{2}} = \frac{y^{2} (\sqrt{x^{2} + y^{2}} - x)}{y^{2}} = \frac{\sqrt{x^{2} + y^{2}} - x}{1} H e n c e, \frac{y^{2}}{\sqrt{x^{2} + y^{2}} + x} = \frac{\sqrt{x^{2} + y^{2}} - x}{1}, w h e r e d e n o m i n a t o r p a r t i s a r a t i o n a l n u m b e r . (3) G i v e n, a = \frac{3 + \sqrt{7}}{2} S o, \frac{1}{a} = \frac{2}{3 + \sqrt{7}} \times \frac{3 - \sqrt{7}}{3 - \sqrt{7}} = \frac{2 (3 - \sqrt{7})}{9 - 7} = 3 - \sqrt{7} N o w, a^{2} + \frac{1}{a^{2}} = {(a + \frac{1}{a})}^{2} - 2 = {(\frac{3 + \sqrt{7}}{2} + 3 - \sqrt{7})}^{2} - 2 = {(\frac{9 - \sqrt{7}}{2})}^{2} - 2 = \frac{81 + 7 - 18 \sqrt{7} - 8}{4} = \frac{40 - 9 \sqrt{7}}{2} H e n c e, a^{2} + \frac{1}{a^{2}} = \frac{40 - 9 \sqrt{7}}{2}$

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