Solve 2sqrt{1/27}-2/3sqrt{18}-sqrt{4/3}+4sqrt{1/2}

$$2 \sqrt{ \frac{ 1 }{ 27 } } - \frac{ 2 }{ 3 } \sqrt{ 18 } - \sqrt{ \frac{ 4 }{ 3 } } +4 \sqrt{ \frac{ 1 }{ 2 } }$$

Evaluate

$$-\frac{4\sqrt{3}}{9}\approx -0.769800359$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{27}} as the division of square roots \frac{\sqrt{1}}{\sqrt{27}}.

$$2\times \frac{\sqrt{1}}{\sqrt{27}}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Calculate the square root of 1 and get 1.

$$2\times \frac{1}{\sqrt{27}}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$2\times \frac{1}{3\sqrt{3}}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

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

$$2\times \frac{\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

The square of \sqrt{3} is 3.

$$2\times \frac{\sqrt{3}}{3\times 3}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Multiply 3 and 3 to get 9.

$$2\times \frac{\sqrt{3}}{9}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Express 2\times \frac{\sqrt{3}}{9} as a single fraction.

$$\frac{2\sqrt{3}}{9}-\frac{2}{3}\sqrt{18}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{2\sqrt{3}}{9}-\frac{2}{3}\times 3\sqrt{2}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Cancel out 3 and 3.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\sqrt{\frac{4}{3}}+4\sqrt{\frac{1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{\sqrt{4}}{\sqrt{3}}+4\sqrt{\frac{1}{2}}$$

Calculate the square root of 4 and get 2.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2}{\sqrt{3}}+4\sqrt{\frac{1}{2}}$$

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

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+4\sqrt{\frac{1}{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2\sqrt{3}}{3}+4\sqrt{\frac{1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2\sqrt{3}}{3}+4\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2\sqrt{3}}{3}+4\times \frac{1}{\sqrt{2}}$$

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

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2\sqrt{3}}{3}+4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{3}}{9}-2\sqrt{2}-\frac{2\sqrt{3}}{3}+4\times \frac{\sqrt{2}}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

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

Combine -2\sqrt{2} and 2\sqrt{2} to get 0.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 3 is 9. Multiply \frac{2\sqrt{3}}{3} times \frac{3}{3}.

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

Since \frac{2\sqrt{3}}{9} and \frac{3\times 2\sqrt{3}}{9} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\sqrt{3}-3\times 2\sqrt{3}.

$$\frac{2\sqrt{3}-6\sqrt{3}}{9}$$

Do the calculations in 2\sqrt{3}-6\sqrt{3}.

$$\frac{-4\sqrt{3}}{9}$$