Solve frac{2sqrt{5}}{3+sqrt{3}}

$$\frac { 2 \sqrt { 5 } } { 3 + \sqrt { 3 } }$$

Evaluate

$$-\frac{\sqrt{15}}{3}+\sqrt{5}\approx 0.945073529$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{5}}{3+\sqrt{3}} by multiplying numerator and denominator by 3-\sqrt{3}.

$$\frac{2\sqrt{5}\left(3-\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}$$

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}.

$$\frac{2\sqrt{5}\left(3-\sqrt{3}\right)}{3^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 3. Square \sqrt{3}.

$$\frac{2\sqrt{5}\left(3-\sqrt{3}\right)}{9-3}$$

Subtract 3 from 9 to get 6.

$$\frac{2\sqrt{5}\left(3-\sqrt{3}\right)}{6}$$

Use the distributive property to multiply 2\sqrt{5} by 3-\sqrt{3}.

$$\frac{6\sqrt{5}-2\sqrt{3}\sqrt{5}}{6}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{6\sqrt{5}-2\sqrt{15}}{6}$$