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

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

Evaluate

$$\frac{47\sqrt{6}}{30}\approx 3.83753393$$

Solution Steps

Multiply 2 and 3 to get 6.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3, the greatest common factor in 6 and 3.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{6} is 6.

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

Combine 2\sqrt{6} and \frac{\sqrt{6}}{6} to get \frac{13}{6}\sqrt{6}.

$$\frac{13}{6}\sqrt{6}-\frac{1}{5}\sqrt{54}$$

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}.

$$\frac{13}{6}\sqrt{6}-\frac{1}{5}\times 3\sqrt{6}$$

Express -\frac{1}{5}\times 3 as a single fraction.

$$\frac{13}{6}\sqrt{6}+\frac{-3}{5}\sqrt{6}$$

Fraction \frac{-3}{5} can be rewritten as -\frac{3}{5} by extracting the negative sign.

$$\frac{13}{6}\sqrt{6}-\frac{3}{5}\sqrt{6}$$

Combine \frac{13}{6}\sqrt{6} and -\frac{3}{5}\sqrt{6} to get \frac{47}{30}\sqrt{6}.

$$\frac{47}{30}\sqrt{6}$$