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

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

Evaluate

$$\frac{\sqrt{2}}{2}+\frac{\sqrt{10}}{2}+\sqrt{5}+4\approx 8.524313589$$

Short Solution Steps

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

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

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

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2}{2}.

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

Since \frac{\left(1+\sqrt{5}\right)\left(2+\sqrt{2}\right)}{2} and \frac{3\times 2}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(1+\sqrt{5}\right)\left(2+\sqrt{2}\right)+3\times 2.

$$\frac{2+\sqrt{2}+2\sqrt{5}+\sqrt{10}+6}{2}$$

Do the calculations in 2+\sqrt{2}+2\sqrt{5}+\sqrt{10}+6.

$$\frac{8+\sqrt{10}+\sqrt{2}+2\sqrt{5}}{2}$$