Solve 2sqrt{2/3}+sqrt{1/6}-1/5sqrt{54}

$$2 \sqrt { \frac { 2 } { 3 } } + \sqrt { \frac { 1 } { 6 } } - \frac { 1 } { 5 \sqrt { 54 } }$$

Evaluate

$$\frac{37\sqrt{6}}{45}\approx 2.0140249$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$2\times \frac{\sqrt{2}}{\sqrt{3}}+\sqrt{\frac{1}{6}}-\frac{1}{5\sqrt{54}}$$

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

$$2\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{\frac{1}{6}}-\frac{1}{5\sqrt{54}}$$

The square of \sqrt{3} is 3.

$$2\times \frac{\sqrt{2}\sqrt{3}}{3}+\sqrt{\frac{1}{6}}-\frac{1}{5\sqrt{54}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2\times \frac{\sqrt{6}}{3}+\sqrt{\frac{1}{6}}-\frac{1}{5\sqrt{54}}$$

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

$$\frac{2\sqrt{6}}{3}+\sqrt{\frac{1}{6}}-\frac{1}{5\sqrt{54}}$$

Rewrite the square root of the division \sqrt{\frac{1}{6}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6}}.

$$\frac{2\sqrt{6}}{3}+\frac{\sqrt{1}}{\sqrt{6}}-\frac{1}{5\sqrt{54}}$$

Calculate the square root of 1 and get 1.

$$\frac{2\sqrt{6}}{3}+\frac{1}{\sqrt{6}}-\frac{1}{5\sqrt{54}}$$

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

$$\frac{2\sqrt{6}}{3}+\frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}-\frac{1}{5\sqrt{54}}$$

The square of \sqrt{6} is 6.

$$\frac{2\sqrt{6}}{3}+\frac{\sqrt{6}}{6}-\frac{1}{5\sqrt{54}}$$

Combine \frac{2\sqrt{6}}{3} and \frac{\sqrt{6}}{6} to get \frac{5}{6}\sqrt{6}.

$$\frac{5}{6}\sqrt{6}-\frac{1}{5\sqrt{54}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$\frac{5}{6}\sqrt{6}-\frac{1}{5\times 3\sqrt{6}}$$

Multiply 5 and 3 to get 15.

$$\frac{5}{6}\sqrt{6}-\frac{1}{15\sqrt{6}}$$

Rationalize the denominator of \frac{1}{15\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{5}{6}\sqrt{6}-\frac{\sqrt{6}}{15\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{5}{6}\sqrt{6}-\frac{\sqrt{6}}{15\times 6}$$

Multiply 15 and 6 to get 90.

$$\frac{5}{6}\sqrt{6}-\frac{\sqrt{6}}{90}$$

Combine \frac{5}{6}\sqrt{6} and -\frac{\sqrt{6}}{90} to get \frac{37}{45}\sqrt{6}.

$$\frac{37}{45}\sqrt{6}$$