Solve left(3-sqrt{3}right)left(1+1/sqrt{3}right)

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

Evaluate

$$2$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

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}.

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 9 to get 6.

$$\frac{6}{3}$$

Divide 6 by 3 to get 2.

$$2$$

Factor

$$2$$