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

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

Evaluate

$$\frac{\sqrt{6}+\sqrt{15}}{3}\approx 2.10749103$$

Solution Steps

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

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

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

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

Square \sqrt{5}. Square \sqrt{2}.

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

Subtract 2 from 5 to get 3.

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

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

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

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

$$\frac{\sqrt{15}+\sqrt{3}\sqrt{2}}{3}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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