Solve frac{3+sqrt{5}}{sqrt{7}-sqrt{3}}

$$\frac { 3 + \sqrt { 5 } } { \sqrt { 7 } - \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{15}+\sqrt{35}+3\sqrt{3}+3\sqrt{7}}{4}\approx 5.730617371$$

Solution Steps

Rationalize the denominator of \frac{3+\sqrt{5}}{\sqrt{7}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{7}+\sqrt{3}.

$$\frac{\left(3+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{3}\right)}{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}+\sqrt{3}\right)}$$

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}.

$$\frac{\left(3+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{3}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(3+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{3}\right)}{7-3}$$

Subtract 3 from 7 to get 4.

$$\frac{\left(3+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{3}\right)}{4}$$

Apply the distributive property by multiplying each term of 3+\sqrt{5} by each term of \sqrt{7}+\sqrt{3}.

$$\frac{3\sqrt{7}+3\sqrt{3}+\sqrt{5}\sqrt{7}+\sqrt{5}\sqrt{3}}{4}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{3\sqrt{7}+3\sqrt{3}+\sqrt{35}+\sqrt{5}\sqrt{3}}{4}$$

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

$$\frac{3\sqrt{7}+3\sqrt{3}+\sqrt{35}+\sqrt{15}}{4}$$