Solve -1/2*1/sqrt{2}-frac{sqrt{3}}{2}-1/sqrt{2}

$$- \frac{ 1 }{ 2 } \times \frac{ 1 }{ \sqrt{ 2 } } - \frac{ \sqrt{ 3 } }{ 2 } - \frac{ 1 }{ \sqrt{ 2 } }$$

Evaluate

$$-\frac{\sqrt{3}}{2}-\frac{3\sqrt{2}}{4}\approx -1.926685576$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

$$-\frac{1}{2}\times \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}$$

Multiply -\frac{1}{2} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\sqrt{2}}{2\times 2}-\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}$$

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

$$\frac{-\sqrt{2}}{2\times 2}-\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-\sqrt{2}}{2\times 2}-\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}$$

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

$$\frac{-\sqrt{2}}{2\times 2}-\frac{2\sqrt{3}}{2\times 2}-\frac{\sqrt{2}}{2}$$

Since \frac{-\sqrt{2}}{2\times 2} and \frac{2\sqrt{3}}{2\times 2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\sqrt{2}-2\sqrt{3}}{2\times 2}-\frac{\sqrt{2}}{2}$$

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

$$\frac{-\sqrt{2}-2\sqrt{3}}{2\times 2}-\frac{2\sqrt{2}}{2\times 2}$$

Since \frac{-\sqrt{2}-2\sqrt{3}}{2\times 2} and \frac{2\sqrt{2}}{2\times 2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\sqrt{2}-2\sqrt{3}-2\sqrt{2}}{2\times 2}$$

Do the calculations in -\sqrt{2}-2\sqrt{3}-2\sqrt{2}.

$$\frac{-3\sqrt{2}-2\sqrt{3}}{2\times 2}$$

Expand 2\times 2.

$$\frac{-3\sqrt{2}-2\sqrt{3}}{4}$$

Factor

$$\frac{-2 \sqrt{3} - 3 \sqrt{2}}{4} = -1.92668557556426$$