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

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

Evaluate

$$4-2\sqrt{3}\approx 0.535898385$$

Short Solution Steps

Anything divided by one gives itself.

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

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

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

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

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

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

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

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

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

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

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

Cancel out 3 in both numerator and denominator.

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

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

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

Consider \left(\sqrt{3}+3\right)\left(\sqrt{3}-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}.

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

Square \sqrt{3}. Square 3.

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

Subtract 9 from 3 to get -6.

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

Cancel out -6, the greatest common factor in 2 and -6.

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

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

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

The square of \sqrt{3} is 3.

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

Combine 3\sqrt{3} and 3\sqrt{3} to get 6\sqrt{3}.

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

Subtract 9 from -3 to get -12.

$$\frac{-12+6\sqrt{3}}{3}$$

Factor

$$2 {(2 - \sqrt{3})} = 0.535898385$$