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

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

Evaluate

$$-\frac{2\sqrt{2}}{3}-6\approx -6.942809042$$

Short Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

Calculate -\sqrt{3} to the power of 2 and get \left(\sqrt{3}\right)^{2}.

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

Express \sqrt{2}\times \frac{\sqrt{2}}{2} as a single fraction.

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

Since \frac{\sqrt{2}\sqrt{2}}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \sqrt{2}\sqrt{2}+1.

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

Do the calculations in 2+1.

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

The square of \sqrt{3} is 3.

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

Multiply 3 and 3 to get 9.

$$\frac{-\sqrt{2}-9}{\frac{3}{2}}$$

Divide -\sqrt{2}-9 by \frac{3}{2} by multiplying -\sqrt{2}-9 by the reciprocal of \frac{3}{2}.

$$\frac{\left(-\sqrt{2}-9\right)\times 2}{3}$$

Use the distributive property to multiply -\sqrt{2}-9 by 2.

$$\frac{2\left(-\sqrt{2}\right)-18}{3}$$

Multiply 2 and -1 to get -2.

$$\frac{-2\sqrt{2}-18}{3}$$