Solve frac{1-frac{sqrt{3}}{3}}{1+frac{sqrt{3}}{2}}

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

Evaluate

$$-\frac{10\sqrt{3}}{3}+6\approx 0.226497308$$

Short Solution Steps

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

Since \frac{2}{2} and \frac{\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

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

Divide \frac{3-\sqrt{3}}{3} by \frac{2+\sqrt{3}}{2} by multiplying \frac{3-\sqrt{3}}{3} by the reciprocal of \frac{2+\sqrt{3}}{2}.

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

Use the distributive property to multiply 3-\sqrt{3} by 2.

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

Use the distributive property to multiply 3 by 2+\sqrt{3}.

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

Rationalize the denominator of \frac{6-2\sqrt{3}}{6+3\sqrt{3}} by multiplying numerator and denominator by 6-3\sqrt{3}.

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

Consider \left(6+3\sqrt{3}\right)\left(6-3\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 6 to the power of 2 and get 36.

$$\frac{\left(6-2\sqrt{3}\right)\left(6-3\sqrt{3}\right)}{36-\left(3\sqrt{3}\right)^{2}}$$

Expand \left(3\sqrt{3}\right)^{2}.

$$\frac{\left(6-2\sqrt{3}\right)\left(6-3\sqrt{3}\right)}{36-3^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(6-2\sqrt{3}\right)\left(6-3\sqrt{3}\right)}{36-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(6-2\sqrt{3}\right)\left(6-3\sqrt{3}\right)}{36-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{\left(6-2\sqrt{3}\right)\left(6-3\sqrt{3}\right)}{36-27}$$

Subtract 27 from 36 to get 9.

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

Apply the distributive property by multiplying each term of 6-2\sqrt{3} by each term of 6-3\sqrt{3}.

$$\frac{36-18\sqrt{3}-12\sqrt{3}+6\left(\sqrt{3}\right)^{2}}{9}$$

Combine -18\sqrt{3} and -12\sqrt{3} to get -30\sqrt{3}.

$$\frac{36-30\sqrt{3}+6\left(\sqrt{3}\right)^{2}}{9}$$

The square of \sqrt{3} is 3.

$$\frac{36-30\sqrt{3}+6\times 3}{9}$$

Multiply 6 and 3 to get 18.

$$\frac{36-30\sqrt{3}+18}{9}$$

Add 36 and 18 to get 54.

$$\frac{54-30\sqrt{3}}{9}$$

Factor

$$\frac{2 {(9 - 5 \sqrt{3})}}{3} = 0.226497308103743$$