Solve ((sqrt{5}+sqrt{3})]+[18/(sqrt{5}-sqrt{3})]?

$$( ( \sqrt { 5 } + \sqrt { 3 } ) ] + [ 18 / ( \sqrt { 5 } - \sqrt { 3 } ) ] ?$$

Evaluate

$$10\left(\sqrt{3}+\sqrt{5}\right)\approx 39.681187851$$

Solution Steps

Rationalize the denominator of \frac{18}{\sqrt{5}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}+\sqrt{3}.

$$\sqrt{5}+\sqrt{3}+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}$$

Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\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}.

$$\sqrt{5}+\sqrt{3}+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{3}.

$$\sqrt{5}+\sqrt{3}+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{5-3}$$

Subtract 3 from 5 to get 2.

$$\sqrt{5}+\sqrt{3}+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{2}$$

Divide 18\left(\sqrt{5}+\sqrt{3}\right) by 2 to get 9\left(\sqrt{5}+\sqrt{3}\right).

$$\sqrt{5}+\sqrt{3}+9\left(\sqrt{5}+\sqrt{3}\right)$$

Use the distributive property to multiply 9 by \sqrt{5}+\sqrt{3}.

$$\sqrt{5}+\sqrt{3}+9\sqrt{5}+9\sqrt{3}$$

Combine \sqrt{5} and 9\sqrt{5} to get 10\sqrt{5}.

$$10\sqrt{5}+\sqrt{3}+9\sqrt{3}$$

Combine \sqrt{3} and 9\sqrt{3} to get 10\sqrt{3}.

$$10\sqrt{5}+10\sqrt{3}$$