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

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

Evaluate

$$\sqrt{10}+2\sqrt{2}\approx 5.990704785$$

Solution Steps

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

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

Consider \left(\sqrt{5}-2\right)\left(\sqrt{5}+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}.

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

Square \sqrt{5}. Square 2.

$$\frac{\sqrt{2}\left(\sqrt{5}+2\right)}{5-4}$$

Subtract 4 from 5 to get 1.

$$\frac{\sqrt{2}\left(\sqrt{5}+2\right)}{1}$$

Anything divided by one gives itself.

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

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

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

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

$$\sqrt{10}+2\sqrt{2}$$