Solve frac{sqrt{5}-sqrt{2}}{3}-frac{2sqrt{2}+5}{2}+frac{sqrt{5}+8sqrt{2}+12}{6}

$$\frac { \sqrt { 5 } - \sqrt { 2 } } { 3 } - \frac { 2 \sqrt { 2 } + 5 } { 2 } + \frac { \sqrt { 5 } + 8 \sqrt { 2 } + 12 } { 6 }$$

Evaluate

$$\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{\sqrt{5}-\sqrt{2}}{3} times \frac{2}{2}. Multiply \frac{2\sqrt{2}+5}{2} times \frac{3}{3}.

$$\frac{2\left(\sqrt{5}-\sqrt{2}\right)}{6}-\frac{3\left(2\sqrt{2}+5\right)}{6}+\frac{\sqrt{5}+8\sqrt{2}+12}{6}$$

Since \frac{2\left(\sqrt{5}-\sqrt{2}\right)}{6} and \frac{3\left(2\sqrt{2}+5\right)}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(\sqrt{5}-\sqrt{2}\right)-3\left(2\sqrt{2}+5\right)}{6}+\frac{\sqrt{5}+8\sqrt{2}+12}{6}$$

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

$$\frac{2\sqrt{5}-2\sqrt{2}-6\sqrt{2}-15}{6}+\frac{\sqrt{5}+8\sqrt{2}+12}{6}$$

Do the calculations in 2\sqrt{5}-2\sqrt{2}-6\sqrt{2}-15.

$$\frac{2\sqrt{5}-8\sqrt{2}-15}{6}+\frac{\sqrt{5}+8\sqrt{2}+12}{6}$$

Since \frac{2\sqrt{5}-8\sqrt{2}-15}{6} and \frac{\sqrt{5}+8\sqrt{2}+12}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{5}-8\sqrt{2}-15+\sqrt{5}+8\sqrt{2}+12}{6}$$

Do the calculations in 2\sqrt{5}-8\sqrt{2}-15+\sqrt{5}+8\sqrt{2}+12.

$$\frac{3\sqrt{5}-3}{6}$$

Factor

$$\frac{\sqrt{5} - 1}{2} = 0.6180339887498949$$