Solve frac{sqrt{2}-sqrt{5}}{sqrt{2}+sqrt{5}}=

$$\frac { \sqrt { 2 } - \sqrt { 5 } } { \sqrt { 2 } + \sqrt { 5 } } =$$

Evaluate

$$\frac{2\sqrt{10}-7}{3}\approx -0.225148227$$

Solution Steps

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

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

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}.

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

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

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

Subtract 5 from 2 to get -3.

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

Multiply \sqrt{2}-\sqrt{5} and \sqrt{2}-\sqrt{5} to get \left(\sqrt{2}-\sqrt{5}\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-\sqrt{5}\right)^{2}.

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

The square of \sqrt{2} is 2.

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

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{2-2\sqrt{10}+\left(\sqrt{5}\right)^{2}}{-3}$$

The square of \sqrt{5} is 5.

$$\frac{2-2\sqrt{10}+5}{-3}$$

Add 2 and 5 to get 7.

$$\frac{7-2\sqrt{10}}{-3}$$

Multiply both numerator and denominator by -1.

$$\frac{-7+2\sqrt{10}}{3}$$