Solve frac{-sqrt{2}-frac{3sqrt{2}}{2}}{frac{3sqrt{2}}{2}-sqrt{2}}

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

Evaluate

$$-5$$

Solution Steps

Combine \frac{3\sqrt{2}}{2} and -\sqrt{2} to get \frac{1}{2}\sqrt{2}.

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

Rationalize the denominator of \frac{-\sqrt{2}-\frac{3\sqrt{2}}{2}}{\frac{1}{2}\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

Anything divided by one gives itself.

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

Use the distributive property to multiply -\sqrt{2}-\frac{3\sqrt{2}}{2} by \sqrt{2}.

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

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

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Cancel out 2 and 2.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-2-3$$

Subtract 3 from -2 to get -5.

$$-5$$

Factor

$$-5$$