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

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

Evaluate

$$\frac{2\sqrt{10}+7}{3}\approx 4.44151844$$

Solution Steps

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

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

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

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

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

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

Subtract 2 from 5 to get 3.

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

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

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

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

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

The square of \sqrt{5} is 5.

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

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

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

The square of \sqrt{2} is 2.

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

Add 5 and 2 to get 7.

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