Simplify. $\newline$ $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

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Q. Simplify.

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\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}

Consider Fractions Separately: We have the expression $(\sqrt{5} + \sqrt{3}) / (\sqrt{5} - \sqrt{3}) + (\sqrt{5} - \sqrt{3}) / (\sqrt{5} + \sqrt{3})$ . To simplify, we will first consider each fraction separately and then add them together.
Multiply by Conjugate: To simplify the fractions, we can multiply the numerator and denominator of each fraction by the conjugate of the denominator. The conjugate of $(\sqrt{5} - \sqrt{3})$ is $(\sqrt{5} + \sqrt{3})$ , and the conjugate of $(\sqrt{5} + \sqrt{3})$ is $(\sqrt{5} - \sqrt{3})$ .
Simplify First Fraction: For the first fraction, multiply the numerator and denominator by the conjugate of the denominator: $\left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right) \cdot \left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right)$ .
Divide Numerator and Denominator: Perform the multiplication in the numerator and denominator: $\newline$ Numerator: $(\sqrt{5} + \sqrt{3}) \times (\sqrt{5} + \sqrt{3}) = 5 + 2\sqrt{15} + 3$ . $\newline$ Denominator: $(\sqrt{5} - \sqrt{3}) \times (\sqrt{5} + \sqrt{3}) = 5 - 3$ .
Simplify Second Fraction: Simplify the numerator and denominator: $\newline$ Numerator: $5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$ . $\newline$ Denominator: $5 - 3 = 2$ . $\newline$ So the first fraction simplifies to $(8 + 2\sqrt{15}) / 2$ .
Divide Numerator and Denominator: Divide each term in the numerator by the denominator: $(8 + 2\sqrt{15}) / 2 = 4 + \sqrt{15}$ .
Add Simplified Fractions: Now, we repeat the process for the second fraction, multiplying the numerator and denominator by the conjugate of the denominator: $\left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right) \times \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right)$ .
Combine Like Terms: Perform the multiplication in the numerator and denominator: $\newline$ Numerator: $(\sqrt{5} - \sqrt{3}) \times (\sqrt{5} - \sqrt{3}) = 5 - 2\sqrt{15} + 3$ . $\newline$ Denominator: $(\sqrt{5} + \sqrt{3}) \times (\sqrt{5} - \sqrt{3}) = 5 - 3$ .
Final Simplified Form: Simplify the numerator and denominator: $\newline$ Numerator: $5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$ . $\newline$ Denominator: $5 - 3 = 2$ . $\newline$ So the second fraction simplifies to $(8 - 2\sqrt{15}) / 2$ .
Final Simplified Form: Simplify the numerator and denominator: $\newline$ Numerator: $5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$ . $\newline$ Denominator: $5 - 3 = 2$ . $\newline$ So the second fraction simplifies to $(8 - 2\sqrt{15}) / 2$ .Divide each term in the numerator by the denominator: $\newline$ $(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}$ .
Final Simplified Form: Simplify the numerator and denominator: $\newline$ Numerator: $5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$ . $\newline$ Denominator: $5 - 3 = 2$ . $\newline$ So the second fraction simplifies to $(8 - 2\sqrt{15}) / 2$ .Divide each term in the numerator by the denominator: $\newline$ $(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}$ .Now, add the simplified forms of the two fractions together: $\newline$ $(4 + \sqrt{15}) + (4 - \sqrt{15})$ .
Final Simplified Form: Simplify the numerator and denominator: $\newline$ Numerator: $5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$ . $\newline$ Denominator: $5 - 3 = 2$ . $\newline$ So the second fraction simplifies to $(8 - 2\sqrt{15}) / 2$ .Divide each term in the numerator by the denominator: $\newline$ $(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}$ .Now, add the simplified forms of the two fractions together: $\newline$ $(4 + \sqrt{15}) + (4 - \sqrt{15})$ .Combine like terms: $\newline$ $4 + \sqrt{15} + 4 - \sqrt{15} = 8$ .
Final Simplified Form: Simplify the numerator and denominator: $\newline$ Numerator: $5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$ . $\newline$ Denominator: $5 - 3 = 2$ . $\newline$ So the second fraction simplifies to $(8 - 2\sqrt{15}) / 2$ .Divide each term in the numerator by the denominator: $\newline$ $(8 - 2\sqrt{15}) / 2 = 4 - \sqrt{15}$ .Now, add the simplified forms of the two fractions together: $\newline$ $(4 + \sqrt{15}) + (4 - \sqrt{15})$ .Combine like terms: $\newline$ $4 + \sqrt{15} + 4 - \sqrt{15} = 8$ .The final simplified form of the expression is $8$ .

Simplify.\newline5+35−3+5−35+3\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}5​−3​5​+3​​+5​+3​5​−3​​

Simplify. $\newline$ $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$