Simplify. $\newline$ $\frac{4}{3+\sqrt{5}}$

Full solution

Q. Simplify.

\newline

\frac{4}{3+\sqrt{5}}

Rationalize Denominator: Rationalize the denominator of the fraction $(\frac{4}{3+\sqrt{5}})$ . To do this, we multiply the numerator and the denominator by the conjugate of the denominator, which is $(3-\sqrt{5})$ . $(\frac{4}{3+\sqrt{5}}) \times (\frac{3-\sqrt{5}}{3-\sqrt{5}})$
Apply Distributive Property: Apply the distributive property (also known as the FOIL method) to the numerator. $\newline$ $(4 \times 3) - (4 \times \sqrt{5}) = 12 - 4\sqrt{5}$
Calculate Denominator Squares: Apply the distributive property to the denominator. $\newline$ $(3+\sqrt{5}) \times (3-\sqrt{5}) = 3^2 - (\sqrt{5})^2$
Subtract Denominator Values: Calculate the squares in the denominator. $\newline$ $3^2 - (\sqrt{5})^2 = 9 - 5$
Write Simplified Expression: Subtract the values in the denominator. $9 - 5 = 4$
Simplify Fraction: Write the simplified expression with the new numerator and denominator. $\newline$ $(12 - 4\sqrt{5}) / 4$
Simplify Fraction: Write the simplified expression with the new numerator and denominator. $\newline$ $(12 - 4\sqrt{5}) / 4$ Simplify the fraction by dividing both terms in the numerator by the denominator. $\newline$ $(12/4) - (4\sqrt{5}/4) = 3 - \sqrt{5}$