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\( x=\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}} \) find \( x^{2}+\frac{1}{x^{2}} \) and \( x^{2}-\frac{1}{x^{2}} \)

Ask by Wheeler Howell. in Pakistan
Mar 18,2025

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Answer

\( x^2 + \frac{1}{x^2} \approx 5.454956 \) and \( x^2 - \frac{1}{x^2} \approx 5.353573 \).

Solution

Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)}\) - step1: Remove the parentheses: \(\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\) - step2: Multiply by the Conjugate: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}\) - step3: Multiply the numbers: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}{3}\) - step4: Multiply the numbers: \(\frac{7+2\sqrt{10}}{3}\) Calculate or simplify the expression \( (\sqrt(5)+\sqrt(2))/(\sqrt(5)-\sqrt(2))^2 + 1/((\sqrt(5)+\sqrt(2))/(\sqrt(5)-\sqrt(2)))^2 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)^{2}}+\frac{1}{\left(\frac{\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)}\right)^{2}}\) - step1: Remove the parentheses: \(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)^{2}}+\frac{1}{\left(\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\right)^{2}}\) - step2: Simplify: \(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)^{2}}+\frac{\left(\sqrt{5}-\sqrt{2}\right)^{2}}{\left(\sqrt{5}+\sqrt{2}\right)^{2}}\) - step3: Evaluate the power: \(\frac{\sqrt{5}+\sqrt{2}}{7-2\sqrt{10}}+\frac{\left(\sqrt{5}-\sqrt{2}\right)^{2}}{\left(\sqrt{5}+\sqrt{2}\right)^{2}}\) - step4: Evaluate the power: \(\frac{\sqrt{5}+\sqrt{2}}{7-2\sqrt{10}}+\frac{7-2\sqrt{10}}{7+2\sqrt{10}}\) - step5: Reduce fractions to a common denominator: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)}{\left(7-2\sqrt{10}\right)\left(7+2\sqrt{10}\right)}+\frac{\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{\left(7+2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}\) - step6: Multiply the terms: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)}{9}+\frac{\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{\left(7+2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}\) - step7: Multiply the terms: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)}{9}+\frac{\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{9}\) - step8: Transform the expression: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)+\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{9}\) - step9: Multiply the terms: \(\frac{11\sqrt{5}+17\sqrt{2}+\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{9}\) - step10: Multiply the terms: \(\frac{11\sqrt{5}+17\sqrt{2}+89-28\sqrt{10}}{9}\) Calculate or simplify the expression \( (\sqrt(5)+\sqrt(2))/(\sqrt(5)-\sqrt(2))^2 - 1/((\sqrt(5)+\sqrt(2))/(\sqrt(5)-\sqrt(2)))^2 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)^{2}}-\frac{1}{\left(\frac{\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}-\sqrt{2}\right)}\right)^{2}}\) - step1: Remove the parentheses: \(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)^{2}}-\frac{1}{\left(\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\right)^{2}}\) - step2: Simplify: \(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)^{2}}-\frac{\left(\sqrt{5}-\sqrt{2}\right)^{2}}{\left(\sqrt{5}+\sqrt{2}\right)^{2}}\) - step3: Evaluate the power: \(\frac{\sqrt{5}+\sqrt{2}}{7-2\sqrt{10}}-\frac{\left(\sqrt{5}-\sqrt{2}\right)^{2}}{\left(\sqrt{5}+\sqrt{2}\right)^{2}}\) - step4: Evaluate the power: \(\frac{\sqrt{5}+\sqrt{2}}{7-2\sqrt{10}}+\frac{-7+2\sqrt{10}}{7+2\sqrt{10}}\) - step5: Rewrite the expression: \(\frac{\sqrt{5}+\sqrt{2}}{7-2\sqrt{10}}-\frac{7-2\sqrt{10}}{7+2\sqrt{10}}\) - step6: Reduce fractions to a common denominator: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)}{\left(7-2\sqrt{10}\right)\left(7+2\sqrt{10}\right)}-\frac{\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{\left(7+2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}\) - step7: Multiply the terms: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)}{9}-\frac{\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{\left(7+2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}\) - step8: Multiply the terms: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)}{9}-\frac{\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{9}\) - step9: Transform the expression: \(\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7+2\sqrt{10}\right)-\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{9}\) - step10: Multiply the terms: \(\frac{11\sqrt{5}+17\sqrt{2}-\left(7-2\sqrt{10}\right)\left(7-2\sqrt{10}\right)}{9}\) - step11: Multiply the terms: \(\frac{11\sqrt{5}+17\sqrt{2}-\left(89-28\sqrt{10}\right)}{9}\) - step12: Remove the parentheses: \(\frac{11\sqrt{5}+17\sqrt{2}-89+28\sqrt{10}}{9}\) To find \( x^2 + \frac{1}{x^2} \) and \( x^2 - \frac{1}{x^2} \) for \( x = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \), we can follow these steps: 1. **Calculate \( x \)**: \[ x = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{7 + 2\sqrt{10}}{3} \quad \text{(exact form)} \] or approximately \( x \approx 4.441518 \). 2. **Calculate \( x^2 \)**: \[ x^2 = \left( \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \right)^2 = \frac{(\sqrt{5} + \sqrt{2})^2}{(\sqrt{5} - \sqrt{2})^2} = \frac{7 + 2\sqrt{10}}{3}^2 \] 3. **Calculate \( \frac{1}{x^2} \)**: \[ \frac{1}{x^2} = \left( \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \right)^2 = \frac{(\sqrt{5} - \sqrt{2})^2}{(\sqrt{5} + \sqrt{2})^2} = \frac{11\sqrt{5} + 17\sqrt{2} - 89 + 28\sqrt{10}}{9} \quad \text{(exact form)} \] or approximately \( \frac{11\sqrt{5} + 17\sqrt{2} - 89 + 28\sqrt{10}}{9} \approx 5.353573 \). 4. **Calculate \( x^2 + \frac{1}{x^2} \)**: \[ x^2 + \frac{1}{x^2} = \frac{11\sqrt{5} + 17\sqrt{2} + 89 - 28\sqrt{10}}{9} \quad \text{(exact form)} \] or approximately \( x^2 + \frac{1}{x^2} \approx 5.454956 \). 5. **Calculate \( x^2 - \frac{1}{x^2} \)**: \[ x^2 - \frac{1}{x^2} = \frac{11\sqrt{5} + 17\sqrt{2} - 89 + 28\sqrt{10}}{9} \quad \text{(exact form)} \] or approximately \( x^2 - \frac{1}{x^2} \approx 5.353573 \). ### Final Results: - \( x^2 + \frac{1}{x^2} = \frac{11\sqrt{5} + 17\sqrt{2} + 89 - 28\sqrt{10}}{9} \) or approximately \( 5.454956 \). - \( x^2 - \frac{1}{x^2} = \frac{11\sqrt{5} + 17\sqrt{2} - 89 + 28\sqrt{10}}{9} \) or approximately \( 5.353573 \).

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The Deep Dive

To find \( x^{2}+\frac{1}{x^{2}} \) and \( x^{2}-\frac{1}{x^{2}} \), we start with \[ x = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}. \] First, simplify \( x \) by multiplying the numerator and denominator by the conjugate of the denominator: \[ x = \frac{(\sqrt{5}+\sqrt{2})(\sqrt{5}+\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})} = \frac{(\sqrt{5}+\sqrt{2})^2}{5 - 2} = \frac{(\sqrt{5}+\sqrt{2})^2}{3}. \] Now, calculate \( (\sqrt{5} + \sqrt{2})^2 \): \[ (\sqrt{5} + \sqrt{2})^2 = 5 + 2 + 2\sqrt{5}\sqrt{2} = 7 + 2\sqrt{10}. \] Thus, \[ x = \frac{7 + 2\sqrt{10}}{3}. \] Next, we compute \( x^2 \): \[ x^2 = \left(\frac{7 + 2\sqrt{10}}{3}\right)^2 = \frac{(7 + 2\sqrt{10})^2}{9} = \frac{49 + 28\sqrt{10} + 40}{9} = \frac{89 + 28\sqrt{10}}{9}. \] Now let's find \( \frac{1}{x} \): \[ \frac{1}{x} = \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}. \] Again, multiply by the conjugate of the denominator: \[ \frac{1}{x} = \frac{(\sqrt{5}-\sqrt{2})(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})} = \frac{(\sqrt{5}-\sqrt{2})^2}{3} = \frac{5 + 2 - 2\sqrt{10}}{3} = \frac{7-2\sqrt{10}}{3}. \] Now calculate \( \frac{1}{x^2} \): \[ \frac{1}{x^2} = \left(\frac{7 - 2\sqrt{10}}{3}\right)^2 = \frac{(7 - 2\sqrt{10})^2}{9} = \frac{49 - 28\sqrt{10} + 40}{9} = \frac{89 - 28\sqrt{10}}{9}. \] Now, we can compute \( x^2 + \frac{1}{x^2} \): \[ x^2 + \frac{1}{x^2} = \frac{89 + 28\sqrt{10}}{9} + \frac{89 - 28\sqrt{10}}{9} = \frac{178}{9}. \] Next, \( x^2 - \frac{1}{x^2} \): \[ x^2 - \frac{1}{x^2} = \frac{89 + 28\sqrt{10}}{9} - \frac{89 - 28\sqrt{10}}{9} = \frac{56\sqrt{10}}{9}. \] The final results are: \[ x^2 + \frac{1}{x^2} = \frac{178}{9}, \quad x^2 - \frac{1}{x^2} = \frac{56\sqrt{10}}{9}. \]

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