Solve 8/sqrt{2+sqrt{5}}

$$\frac { 8 } { \sqrt { 2 } + \sqrt { 5 } }$$

Evaluate

$$\frac{8\left(\sqrt{5}-\sqrt{2}\right)}{3}\approx 2.191611774$$

Solution Steps

Rationalize the denominator of \frac{8}{\sqrt{2}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{5}.

$$\frac{8\left(\sqrt{2}-\sqrt{5}\right)}{\left(\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{5}\right)}$$

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

$$\frac{8\left(\sqrt{2}-\sqrt{5}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

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

$$\frac{8\left(\sqrt{2}-\sqrt{5}\right)}{2-5}$$

Subtract 5 from 2 to get -3.

$$\frac{8\left(\sqrt{2}-\sqrt{5}\right)}{-3}$$

Use the distributive property to multiply 8 by \sqrt{2}-\sqrt{5}.

$$\frac{8\sqrt{2}-8\sqrt{5}}{-3}$$