Solve frac{sqrt{3}}{3}left(x-8/3right)left(x+2/3right)

$$\frac{ \sqrt{ 3 } }{ 3 } \left( x- \frac{ 8 }{ 3 } \right) \left( x+ \frac{ 2 }{ 3 } \right)$$

Evaluate

$$\frac{\sqrt{3}\left(3x-8\right)\left(3x+2\right)}{27}$$

Short Solution Steps

Use the distributive property to multiply \frac{\sqrt{3}}{3} by x-\frac{8}{3}.

$$\left(\frac{\sqrt{3}}{3}x+\frac{\sqrt{3}}{3}\left(-\frac{8}{3}\right)\right)\left(x+\frac{2}{3}\right)$$

Express \frac{\sqrt{3}}{3}x as a single fraction.

$$\left(\frac{\sqrt{3}x}{3}+\frac{\sqrt{3}}{3}\left(-\frac{8}{3}\right)\right)\left(x+\frac{2}{3}\right)$$

Multiply \frac{\sqrt{3}}{3} times -\frac{8}{3} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{\sqrt{3}x}{3}+\frac{-\sqrt{3}\times 8}{3\times 3}\right)\left(x+\frac{2}{3}\right)$$

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

$$\left(\frac{3\sqrt{3}x}{3\times 3}+\frac{-\sqrt{3}\times 8}{3\times 3}\right)\left(x+\frac{2}{3}\right)$$

Since \frac{3\sqrt{3}x}{3\times 3} and \frac{-\sqrt{3}\times 8}{3\times 3} have the same denominator, add them by adding their numerators.

$$\frac{3\sqrt{3}x-\sqrt{3}\times 8}{3\times 3}\left(x+\frac{2}{3}\right)$$

Do the multiplications in 3\sqrt{3}x-\sqrt{3}\times 8.

$$\frac{3\sqrt{3}x-8\sqrt{3}}{3\times 3}\left(x+\frac{2}{3}\right)$$

Factor the expressions that are not already factored in \frac{3\sqrt{3}x-8\sqrt{3}}{3\times 3}.

$$\frac{\sqrt{3}\left(3x-8\right)}{3\times 3}\left(x+\frac{2}{3}\right)$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{3x-8}{3\sqrt{3}}\left(x+\frac{2}{3}\right)$$

Use the distributive property to multiply \frac{3x-8}{3\sqrt{3}} by x+\frac{2}{3}.

$$\frac{3x-8}{3\sqrt{3}}x+\frac{3x-8}{3\sqrt{3}}\times \frac{2}{3}$$

Rationalize the denominator of \frac{3x-8}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(3x-8\right)\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}x+\frac{3x-8}{3\sqrt{3}}\times \frac{2}{3}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3x-8\right)\sqrt{3}}{3\times 3}x+\frac{3x-8}{3\sqrt{3}}\times \frac{2}{3}$$

Multiply 3 and 3 to get 9.

$$\frac{\left(3x-8\right)\sqrt{3}}{9}x+\frac{3x-8}{3\sqrt{3}}\times \frac{2}{3}$$

Express \frac{\left(3x-8\right)\sqrt{3}}{9}x as a single fraction.

$$\frac{\left(3x-8\right)\sqrt{3}x}{9}+\frac{3x-8}{3\sqrt{3}}\times \frac{2}{3}$$

Rationalize the denominator of \frac{3x-8}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(3x-8\right)\sqrt{3}x}{9}+\frac{\left(3x-8\right)\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}\times \frac{2}{3}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3x-8\right)\sqrt{3}x}{9}+\frac{\left(3x-8\right)\sqrt{3}}{3\times 3}\times \frac{2}{3}$$

Multiply 3 and 3 to get 9.

$$\frac{\left(3x-8\right)\sqrt{3}x}{9}+\frac{\left(3x-8\right)\sqrt{3}}{9}\times \frac{2}{3}$$

Multiply \frac{\left(3x-8\right)\sqrt{3}}{9} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3x-8\right)\sqrt{3}x}{9}+\frac{\left(3x-8\right)\sqrt{3}\times 2}{9\times 3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 9\times 3 is 3\times 9. Multiply \frac{\left(3x-8\right)\sqrt{3}x}{9} times \frac{3}{3}.

$$\frac{3\left(3x-8\right)\sqrt{3}x}{3\times 9}+\frac{\left(3x-8\right)\sqrt{3}\times 2}{3\times 9}$$

Since \frac{3\left(3x-8\right)\sqrt{3}x}{3\times 9} and \frac{\left(3x-8\right)\sqrt{3}\times 2}{3\times 9} have the same denominator, add them by adding their numerators.

$$\frac{3\left(3x-8\right)\sqrt{3}x+\left(3x-8\right)\sqrt{3}\times 2}{3\times 9}$$

Do the multiplications in 3\left(3x-8\right)\sqrt{3}x+\left(3x-8\right)\sqrt{3}\times 2.

$$\frac{9x^{2}\sqrt{3}-24\sqrt{3}x+6x\sqrt{3}-16\sqrt{3}}{3\times 9}$$

Combine like terms in 9x^{2}\sqrt{3}-24\sqrt{3}x+6x\sqrt{3}-16\sqrt{3}.

$$\frac{-16\sqrt{3}+9x^{2}\sqrt{3}-18\sqrt{3}x}{3\times 9}$$

Factor the expressions that are not already factored in \frac{-16\sqrt{3}+9x^{2}\sqrt{3}-18\sqrt{3}x}{3\times 9}.

$$\frac{\sqrt{3}\left(3x-8\right)\left(3x+2\right)}{3\times 9}$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{\left(3x-8\right)\left(3x+2\right)}{9\sqrt{3}}$$

Rationalize the denominator of \frac{\left(3x-8\right)\left(3x+2\right)}{9\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(3x-8\right)\left(3x+2\right)\sqrt{3}}{9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3x-8\right)\left(3x+2\right)\sqrt{3}}{9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{\left(3x-8\right)\left(3x+2\right)\sqrt{3}}{27}$$

Apply the distributive property by multiplying each term of 3x-8 by each term of 3x+2.

$$\frac{\left(9x^{2}+6x-24x-16\right)\sqrt{3}}{27}$$

Combine 6x and -24x to get -18x.

$$\frac{\left(9x^{2}-18x-16\right)\sqrt{3}}{27}$$

Use the distributive property to multiply 9x^{2}-18x-16 by \sqrt{3}.

$$\frac{9x^{2}\sqrt{3}-18x\sqrt{3}-16\sqrt{3}}{27}$$

Factor

$$\frac{\sqrt{3}\left(3x-8\right)\left(3x+2\right)}{27}$$