Solve 4(frac{sqrt{2}}{3})^2-3(1/3)^3-(sqrt{3})-(sqrt{3})

$$4 ( \frac { \sqrt { 2 } } { 3 } ) ^ { 2 } - 3 ( \frac { 1 } { 3 } ) ^ { 3 } - ( \sqrt { 3 } ) - ( \sqrt { 3 } )$$

Evaluate

$$\frac{7}{9}-2\sqrt{3}\approx -2.686323837$$

Solution Steps

To raise \frac{\sqrt{2}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Express 4\times \frac{\left(\sqrt{2}\right)^{2}}{3^{2}} as a single fraction.

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

Calculate \frac{1}{3} to the power of 3 and get \frac{1}{27}.

$$\frac{4\left(\sqrt{2}\right)^{2}}{3^{2}}-3\times \frac{1}{27}-\sqrt{3}-\sqrt{3}$$

Multiply 3 and \frac{1}{27} to get \frac{1}{9}.

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

To add or subtract expressions, expand them to make their denominators the same. Expand 3^{2}.

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

Since \frac{4\left(\sqrt{2}\right)^{2}}{9} and \frac{1}{9} have the same denominator, subtract them by subtracting their numerators.

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

Combine -\sqrt{3} and -\sqrt{3} to get -2\sqrt{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{3} times \frac{9}{9}.

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

Since \frac{4\left(\sqrt{2}\right)^{2}-1}{9} and \frac{9\left(-2\right)\sqrt{3}}{9} have the same denominator, add them by adding their numerators.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$\frac{8-1}{9}-2\sqrt{3}$$

Subtract 1 from 8 to get 7.

$$\frac{7}{9}-2\sqrt{3}$$

Factor

$$\frac{7 - 18 \sqrt{3}}{9} = -2.6863238373599767$$