Solve frac{2+sqrt{3}}{1divsqrt{5}}+frac{2-sqrt{3}}{1+sqrt{5}}

$$\frac { 2 + \sqrt { 3 } } { 1 \div \sqrt { 5 } } + \frac { 2 - \sqrt { 3 } } { 1 + \sqrt { 5 } }$$

Evaluate

$$\frac{\sqrt{3}+3\sqrt{15}+10\sqrt{5}-2}{4}\approx 8.427920155$$

Short Solution Steps

Divide 2+\sqrt{3} by \frac{1}{\sqrt{5}} by multiplying 2+\sqrt{3} by the reciprocal of \frac{1}{\sqrt{5}}.

$$\left(2+\sqrt{3}\right)\sqrt{5}+\frac{2-\sqrt{3}}{1+\sqrt{5}}$$

Use the distributive property to multiply 2+\sqrt{3} by \sqrt{5}.

$$2\sqrt{5}+\sqrt{3}\sqrt{5}+\frac{2-\sqrt{3}}{1+\sqrt{5}}$$

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

$$2\sqrt{5}+\sqrt{15}+\frac{2-\sqrt{3}}{1+\sqrt{5}}$$

Rationalize the denominator of \frac{2-\sqrt{3}}{1+\sqrt{5}} by multiplying numerator and denominator by 1-\sqrt{5}.

$$2\sqrt{5}+\sqrt{15}+\frac{\left(2-\sqrt{3}\right)\left(1-\sqrt{5}\right)}{\left(1+\sqrt{5}\right)\left(1-\sqrt{5}\right)}$$

Consider \left(1+\sqrt{5}\right)\left(1-\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}.

$$2\sqrt{5}+\sqrt{15}+\frac{\left(2-\sqrt{3}\right)\left(1-\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 1. Square \sqrt{5}.

$$2\sqrt{5}+\sqrt{15}+\frac{\left(2-\sqrt{3}\right)\left(1-\sqrt{5}\right)}{1-5}$$

Subtract 5 from 1 to get -4.

$$2\sqrt{5}+\sqrt{15}+\frac{\left(2-\sqrt{3}\right)\left(1-\sqrt{5}\right)}{-4}$$

Apply the distributive property by multiplying each term of 2-\sqrt{3} by each term of 1-\sqrt{5}.

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

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

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

Multiply both numerator and denominator by -1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{5}+\sqrt{15} times \frac{4}{4}.

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

Since \frac{4\left(2\sqrt{5}+\sqrt{15}\right)}{4} and \frac{-2+2\sqrt{5}+\sqrt{3}-\sqrt{15}}{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4\left(2\sqrt{5}+\sqrt{15}\right)-2+2\sqrt{5}+\sqrt{3}-\sqrt{15}.

$$\frac{8\sqrt{5}+4\sqrt{15}-2+2\sqrt{5}+\sqrt{3}-\sqrt{15}}{4}$$

Do the calculations in 8\sqrt{5}+4\sqrt{15}-2+2\sqrt{5}+\sqrt{3}-\sqrt{15}.

$$\frac{10\sqrt{5}+3\sqrt{15}-2+\sqrt{3}}{4}$$