Solve frac{sqrt{3}}{5-sqrt{19}}

$$\frac { \sqrt { 3 } } { 5 - \sqrt { 19 } }$$

Evaluate

$$\frac{\sqrt{3}\left(\sqrt{19}+5\right)}{6}\approx 2.701681412$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{5-\sqrt{19}} by multiplying numerator and denominator by 5+\sqrt{19}.

$$\frac{\sqrt{3}\left(5+\sqrt{19}\right)}{\left(5-\sqrt{19}\right)\left(5+\sqrt{19}\right)}$$

Consider \left(5-\sqrt{19}\right)\left(5+\sqrt{19}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{3}\left(5+\sqrt{19}\right)}{5^{2}-\left(\sqrt{19}\right)^{2}}$$

Square 5. Square \sqrt{19}.

$$\frac{\sqrt{3}\left(5+\sqrt{19}\right)}{25-19}$$

Subtract 19 from 25 to get 6.

$$\frac{\sqrt{3}\left(5+\sqrt{19}\right)}{6}$$

Use the distributive property to multiply \sqrt{3} by 5+\sqrt{19}.

$$\frac{5\sqrt{3}+\sqrt{3}\sqrt{19}}{6}$$

To multiply \sqrt{3} and \sqrt{19}, multiply the numbers under the square root.

$$\frac{5\sqrt{3}+\sqrt{57}}{6}$$