Solve 1/3sqrt{2}-2sqrt{2}+7sqrt{2}-1/6sqrt{2}

$$\frac { 1 } { 3 \sqrt { 2 } } - 2 \sqrt { 2 } + 7 \sqrt { 2 } - \frac { 1 } { 6 \sqrt { 2 } }$$

Evaluate

$$\frac{61\sqrt{2}}{12}\approx 7.188918942$$

Solution Steps

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

$$\frac{\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}-2\sqrt{2}+7\sqrt{2}-\frac{1}{6\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{3\times 2}-2\sqrt{2}+7\sqrt{2}-\frac{1}{6\sqrt{2}}$$

Multiply 3 and 2 to get 6.

$$\frac{\sqrt{2}}{6}-2\sqrt{2}+7\sqrt{2}-\frac{1}{6\sqrt{2}}$$

Combine \frac{\sqrt{2}}{6} and -2\sqrt{2} to get -\frac{11}{6}\sqrt{2}.

$$-\frac{11}{6}\sqrt{2}+7\sqrt{2}-\frac{1}{6\sqrt{2}}$$

Combine -\frac{11}{6}\sqrt{2} and 7\sqrt{2} to get \frac{31}{6}\sqrt{2}.

$$\frac{31}{6}\sqrt{2}-\frac{1}{6\sqrt{2}}$$

Rationalize the denominator of \frac{1}{6\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{31}{6}\sqrt{2}-\frac{\sqrt{2}}{6\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{31}{6}\sqrt{2}-\frac{\sqrt{2}}{6\times 2}$$

Multiply 6 and 2 to get 12.

$$\frac{31}{6}\sqrt{2}-\frac{\sqrt{2}}{12}$$

Combine \frac{31}{6}\sqrt{2} and -\frac{\sqrt{2}}{12} to get \frac{61}{12}\sqrt{2}.

$$\frac{61}{12}\sqrt{2}$$