Solve 1/2sqrt{2+sqrt{3}}

$$\frac { 1 } { 2 \sqrt { 2 } + \sqrt { 3 } }$$

Evaluate

$$\frac{2\sqrt{2}-\sqrt{3}}{5}\approx 0.219275263$$

Solution Steps

Rationalize the denominator of \frac{1}{2\sqrt{2}+\sqrt{3}} by multiplying numerator and denominator by 2\sqrt{2}-\sqrt{3}.

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

Consider \left(2\sqrt{2}+\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{2}-\sqrt{3}}{4\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}-\sqrt{3}}{4\times 2-\left(\sqrt{3}\right)^{2}}$$

Multiply 4 and 2 to get 8.

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 8 to get 5.

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