Question

Evaluate 1/3+√7+1/√7+√5+1/√5+√3+1/√3+1. I beg u plz answer I request u also​

Answer 1

We're asked to evaluate:

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$\sf \dashrightarrow \ \dfrac{1}{3 + \sqrt{7}} \ + \ \dfrac{1}{\sqrt{7} + \sqrt{5}} \ + \ \dfrac{1}{\sqrt{5} + \sqrt{3}} \ + \ \dfrac{1}{\sqrt{3} + 1}$

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In order to evaluate the given expression, let's first rationalize the denominators of each fraction in the expression.

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In order to rationalize the denominator of each fraction, we'll need to multiply both the numerator and the denominator of the fraction by the conjugate of the expression in the denominator.

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• Conjugate of 3 + √7 is 3 - √7
• Conjugate of √7 + √5 is √7 - √5
• Conjugate of √5 + √3 is √5 - √3
• Conjugate of √3 + √1 is √3 - √1

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Therefore;

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$\sf \dashrightarrow \ \Bigg[ \ \dfrac{1}{3 + \sqrt{7}} \times \dfrac{3 - \sqrt{7}}{3 - \sqrt{7}} \ \Bigg] + \ \Bigg[ \ \dfrac{1}{\sqrt{7} + \sqrt{5}} \times \dfrac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} \ \Bigg] \\ \\ \\ \\{ \ \ \ \ \ \ \ \ \ + \ \Bigg[ \ \dfrac{1}{\sqrt{5} + \sqrt{3}} \times \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} \ \Bigg] + \ \Bigg[ \ \dfrac{1}{\sqrt{3} + 1} \times \dfrac{\sqrt{3} - 1}{\sqrt{3} - 1} \ \Bigg]}$

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[Broken into two lines for readability]

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Using (a + b)(a - b) = a² - b² in the denominator we get;

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$\sf \dashrightarrow \ \Bigg[\dfrac{3 - \sqrt{7}}{(3)^2 - (\sqrt{7})^2}\Bigg] + \Bigg[\dfrac{\sqrt{7} - \sqrt{5}}{(\sqrt{7})^2 - (\sqrt{5})^2}\Bigg] \\ \\ \\ \\ {\ \ \ \ \ \ \ \ \ \ \ \ \ \ + \ \Bigg[\dfrac{\sqrt{5} - \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2}\Bigg] + \Bigg[\dfrac{\sqrt{3} - 1}{(\sqrt{3})^2 - (1)^2}\Bigg]}$

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$\sf \dashrightarrow \ \Bigg[\dfrac{3 - \sqrt{7}}{9 - 7}\Bigg] \ + \ \Bigg[\dfrac{\sqrt{7} - \sqrt{5}}{7 - 5}\Bigg] \ + \ \Bigg[\dfrac{\sqrt{5} - \sqrt{3}}{5 - 3}\Bigg] \ + \ \Bigg[\dfrac{\sqrt{3} - 1}{3 - 1}\Bigg]$

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$\sf \dashrightarrow \ \dfrac{3 - \sqrt{7}}{2} \ + \ \dfrac{\sqrt{7} - \sqrt{5}}{2} \ + \ \dfrac{\sqrt{5} - \sqrt{3}}{2} \ + \ \dfrac{\sqrt{3} - 1}{2}$

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$\sf \dashrightarrow \ \dfrac{3 - \sqrt{7} + \sqrt{7} - \sqrt{5} + \sqrt{5} - \sqrt{3} + \sqrt{3} - 1}{2}$

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$\dashrightarrow \ \sf \dfrac{3 - 1}{2}$

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$\dashrightarrow \ \sf \dfrac{2}{2}$

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$\dashrightarrow \ \sf 1$

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Therefore, the answer is 1.