Solve -sqrt{3}/(3+sqrt{3})

$$- \sqrt { 3 } / ( 3 + \sqrt { 3 } )$$

Evaluate

$$\frac{1-\sqrt{3}}{2}\approx -0.366025404$$

Solution Steps

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

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

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}.

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

Square 3. Square \sqrt{3}.

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

Subtract 3 from 9 to get 6.

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

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

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

Multiply -1 and -1 to get 1.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Multiply 3 and -1 to get -3.

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