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

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

Evaluate

$$-\frac{\sqrt{3}}{3}+2\approx 1.422649731$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Rationalize the denominator of \frac{1}{3+\sqrt{3}} by multiplying numerator and denominator by 3-\sqrt{3}.

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

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

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

Square 3. Square \sqrt{3}.

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

Subtract 3 from 9 to get 6.

$$3-\frac{\sqrt{3}}{3}-\frac{3-\sqrt{3}}{6}-\frac{1}{3-\sqrt{3}}$$

Rationalize the denominator of \frac{1}{3-\sqrt{3}} by multiplying numerator and denominator by 3+\sqrt{3}.

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

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

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

Square 3. Square \sqrt{3}.

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

Subtract 3 from 9 to get 6.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 6 is 6. Multiply \frac{9-\sqrt{3}}{3} times \frac{2}{2}.

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

Since \frac{2\left(9-\sqrt{3}\right)}{6} and \frac{3-\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(9-\sqrt{3}\right)-\left(3-\sqrt{3}\right).

$$\frac{18-2\sqrt{3}-3+\sqrt{3}}{6}-\frac{3+\sqrt{3}}{6}$$

Do the calculations in 18-2\sqrt{3}-3+\sqrt{3}.

$$\frac{15-\sqrt{3}}{6}-\frac{3+\sqrt{3}}{6}$$

Since \frac{15-\sqrt{3}}{6} and \frac{3+\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 15-\sqrt{3}-\left(3+\sqrt{3}\right).

$$\frac{15-\sqrt{3}-3-\sqrt{3}}{6}$$

Do the calculations in 15-\sqrt{3}-3-\sqrt{3}.

$$\frac{12-2\sqrt{3}}{6}$$