Solve 5+1/sqrt{5}+1/5+sqrt{5}-1/5-sqrt{5}

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

Evaluate

$$\frac{\sqrt{5}}{10}+5\approx 5.223606798$$

Short Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

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

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

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

Square 5. Square \sqrt{5}.

$$5+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{25-5}-\frac{1}{5-\sqrt{5}}$$

Subtract 5 from 25 to get 20.

$$5+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{1}{5-\sqrt{5}}$$

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

$$5+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{\left(5-\sqrt{5}\right)\left(5+\sqrt{5}\right)}$$

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

$$5+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{5^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 5. Square \sqrt{5}.

$$5+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{25-5}$$

Subtract 5 from 25 to get 20.

$$5+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

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

$$\frac{5\times 5}{5}+\frac{\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

Since \frac{5\times 5}{5} and \frac{\sqrt{5}}{5} have the same denominator, add them by adding their numerators.

$$\frac{5\times 5+\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

Do the multiplications in 5\times 5+\sqrt{5}.

$$\frac{25+\sqrt{5}}{5}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 20 is 20. Multiply \frac{25+\sqrt{5}}{5} times \frac{4}{4}.

$$\frac{4\left(25+\sqrt{5}\right)}{20}+\frac{5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

Since \frac{4\left(25+\sqrt{5}\right)}{20} and \frac{5-\sqrt{5}}{20} have the same denominator, add them by adding their numerators.

$$\frac{4\left(25+\sqrt{5}\right)+5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

Do the multiplications in 4\left(25+\sqrt{5}\right)+5-\sqrt{5}.

$$\frac{100+4\sqrt{5}+5-\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

Do the calculations in 100+4\sqrt{5}+5-\sqrt{5}.

$$\frac{105+3\sqrt{5}}{20}-\frac{5+\sqrt{5}}{20}$$

Since \frac{105+3\sqrt{5}}{20} and \frac{5+\sqrt{5}}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{105+3\sqrt{5}-\left(5+\sqrt{5}\right)}{20}$$

Do the multiplications in 105+3\sqrt{5}-\left(5+\sqrt{5}\right).

$$\frac{105+3\sqrt{5}-5-\sqrt{5}}{20}$$

Do the calculations in 105+3\sqrt{5}-5-\sqrt{5}.

$$\frac{100+2\sqrt{5}}{20}$$