Solve frac{-frac{sqrt{2}}{2}*frac{sqrt{3}}{2}+frac{sqrt{2}}{2}*(-sqrt{3})}{-1*frac{sqrt{3}}{2}}

$$\frac { - \frac { \sqrt { 2 } } { 2 } \cdot \frac { \sqrt { 3 } } { 2 } + \frac { \sqrt { 2 } } { 2 } \cdot ( - \sqrt { 3 } ) } { - 1 \cdot \frac { \sqrt { 3 } } { 2 } }$$

Evaluate

$$\frac{3\sqrt{2}}{2}\approx 2.121320344$$

Short Solution Steps

Multiply -\frac{\sqrt{2}}{2} times \frac{\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Express \frac{\sqrt{2}}{2}\left(-\sqrt{3}\right) as a single fraction.

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

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}\sqrt{3}}{2} times \frac{2}{2}.

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

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

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

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

$$\frac{\frac{-\sqrt{6}-2\sqrt{6}}{2\times 2}}{-\frac{\sqrt{3}}{2}}$$

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

$$\frac{\frac{-3\sqrt{6}}{2\times 2}}{-\frac{\sqrt{3}}{2}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{-3\sqrt{6}}{4}}{-\frac{\sqrt{3}}{2}}$$

Express \frac{\frac{-3\sqrt{6}}{4}}{-\frac{\sqrt{3}}{2}} as a single fraction.

$$\frac{-3\sqrt{6}}{4\left(-1\right)\times \frac{\sqrt{3}}{2}}$$

Multiply 4 and -1 to get -4.

$$\frac{-3\sqrt{6}}{-4\times \frac{\sqrt{3}}{2}}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{-3\sqrt{6}}{-2\sqrt{3}}$$

Rationalize the denominator of \frac{-3\sqrt{6}}{-2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{-3\sqrt{6}\sqrt{3}}{-2\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{-3\sqrt{6}\sqrt{3}}{-2\times 3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Multiply -2 and 3 to get -6.

$$\frac{-3\times 3\sqrt{2}}{-6}$$

Multiply -3 and 3 to get -9.

$$\frac{-9\sqrt{2}}{-6}$$

Divide -9\sqrt{2} by -6 to get \frac{3}{2}\sqrt{2}.

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