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

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

Evaluate

$$-\frac{\sqrt{15}}{2}+2\sqrt{5}+\frac{5}{2}-2\sqrt{3}\approx 1.571542667$$

Solution Steps

Rationalize the denominator of \frac{4+\sqrt{5}}{\sqrt{3}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{5}.

$$\frac{\left(4+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)}{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)}$$

Consider \left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(4+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)}{\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Square \sqrt{3}. Square \sqrt{5}.

$$\frac{\left(4+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)}{3-5}$$

Subtract 5 from 3 to get -2.

$$\frac{\left(4+\sqrt{5}\right)\left(\sqrt{3}-\sqrt{5}\right)}{-2}$$

Apply the distributive property by multiplying each term of 4+\sqrt{5} by each term of \sqrt{3}-\sqrt{5}.

$$\frac{4\sqrt{3}-4\sqrt{5}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}}{-2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{4\sqrt{3}-4\sqrt{5}+\sqrt{15}-\left(\sqrt{5}\right)^{2}}{-2}$$

The square of \sqrt{5} is 5.

$$\frac{4\sqrt{3}-4\sqrt{5}+\sqrt{15}-5}{-2}$$

Multiply both numerator and denominator by -1.

$$\frac{-4\sqrt{3}+4\sqrt{5}-\sqrt{15}+5}{2}$$