Solve frac{sqrt{3}}{sqrt{7}-sqrt{3}}

$$\frac { \sqrt { 3 } } { \sqrt { 7 } - \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{21}+3}{4}\approx 1.895643924$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{7}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{7}+\sqrt{3}.

$$\frac{\sqrt{3}\left(\sqrt{7}+\sqrt{3}\right)}{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}+\sqrt{3}\right)}$$

Consider \left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}+\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}.

$$\frac{\sqrt{3}\left(\sqrt{7}+\sqrt{3}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \sqrt{7}. Square \sqrt{3}.

$$\frac{\sqrt{3}\left(\sqrt{7}+\sqrt{3}\right)}{7-3}$$

Subtract 3 from 7 to get 4.

$$\frac{\sqrt{3}\left(\sqrt{7}+\sqrt{3}\right)}{4}$$

Use the distributive property to multiply \sqrt{3} by \sqrt{7}+\sqrt{3}.

$$\frac{\sqrt{3}\sqrt{7}+\left(\sqrt{3}\right)^{2}}{4}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{21}+\left(\sqrt{3}\right)^{2}}{4}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{21}+3}{4}$$