Solve (frac{3+sqrt{6}}{3})^3-3(frac{3+sqrt{6}}{3})^2+frac{3sqrt{6}}{3}+1

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

Evaluate

$$\frac{2\sqrt{6}}{9}-1\approx -0.455668946$$

Short Solution Steps

To raise \frac{3+\sqrt{6}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{3+\sqrt{6}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Express 3\times \frac{\left(3+\sqrt{6}\right)^{2}}{3^{2}} as a single fraction.

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

Cancel out 3 in both numerator and denominator.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{6}+3\right)^{2}.

$$\frac{\left(3+\sqrt{6}\right)^{3}}{3^{3}}-\frac{\left(\sqrt{6}\right)^{2}+6\sqrt{6}+9}{3}+\frac{3\sqrt{6}}{3}+1$$

The square of \sqrt{6} is 6.

$$\frac{\left(3+\sqrt{6}\right)^{3}}{3^{3}}-\frac{6+6\sqrt{6}+9}{3}+\frac{3\sqrt{6}}{3}+1$$

Add 6 and 9 to get 15.

$$\frac{\left(3+\sqrt{6}\right)^{3}}{3^{3}}-\frac{15+6\sqrt{6}}{3}+\frac{3\sqrt{6}}{3}+1$$

Divide each term of 15+6\sqrt{6} by 3 to get 5+2\sqrt{6}.

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

To find the opposite of 5+2\sqrt{6}, find the opposite of each term.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -5-2\sqrt{6} times \frac{3^{3}}{3^{3}}.

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

Since \frac{\left(3+\sqrt{6}\right)^{3}}{3^{3}} and \frac{\left(-5-2\sqrt{6}\right)\times 3^{3}}{3^{3}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3+\sqrt{6}\right)^{3}+\left(-5-2\sqrt{6}\right)\times 3^{3}.

$$\frac{3^{3}+3\times 3^{2}\sqrt{6}+3\times 3\left(\sqrt{6}\right)^{2}+\left(\sqrt{6}\right)^{3}-135-54\sqrt{6}}{3^{3}}+\frac{3\sqrt{6}}{3}+1$$

Do the calculations in 3^{3}+3\times 3^{2}\sqrt{6}+3\times 3\left(\sqrt{6}\right)^{2}+\left(\sqrt{6}\right)^{3}-135-54\sqrt{6}.

$$\frac{-54-21\sqrt{6}}{3^{3}}+\frac{3\sqrt{6}}{3}+1$$

Cancel out 3 and 3.

$$\frac{-54-21\sqrt{6}}{3^{3}}+\sqrt{6}+1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{6}+1 times \frac{3^{3}}{3^{3}}.

$$\frac{-54-21\sqrt{6}}{3^{3}}+\frac{\left(\sqrt{6}+1\right)\times 3^{3}}{3^{3}}$$

Since \frac{-54-21\sqrt{6}}{3^{3}} and \frac{\left(\sqrt{6}+1\right)\times 3^{3}}{3^{3}} have the same denominator, add them by adding their numerators.

$$\frac{-54-21\sqrt{6}+\left(\sqrt{6}+1\right)\times 3^{3}}{3^{3}}$$

Do the multiplications in -54-21\sqrt{6}+\left(\sqrt{6}+1\right)\times 3^{3}.

$$\frac{-54-21\sqrt{6}+27\sqrt{6}+27}{3^{3}}$$

Do the calculations in -54-21\sqrt{6}+27\sqrt{6}+27.

$$\frac{-27+6\sqrt{6}}{3^{3}}$$

Expand 3^{3}.

$$\frac{-27+6\sqrt{6}}{27}$$

Factor

$$\frac{2 \sqrt{6} - 9}{9} = -0.4556689460481827$$