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

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

Evaluate

$$\frac{\sqrt{2}-4}{6}\approx -0.430964406$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply \frac{\sqrt{2}}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

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

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

Expand 2\times 3.

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

Factor

$$\frac{\sqrt{2} - 4}{6} = -0.43096440627115085$$