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

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

Evaluate

$$\frac{4-\sqrt{10}}{3}\approx 0.27924078$$

Solution Steps

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

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

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

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

Expand \left(2\sqrt{2}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

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

The square of \sqrt{5} is 5.

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

Subtract 5 from 8 to get 3.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

$$\frac{4-\sqrt{2}\sqrt{5}}{3}$$

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

$$\frac{4-\sqrt{10}}{3}$$