Solve 1/2sqrt{2+sqrt{7}}

$$\frac { 1 } { 2 \sqrt { 2 } + \sqrt { 7 } }$$

Evaluate

$$2\sqrt{2}-\sqrt{7}\approx 0.182675814$$

Solution Steps

Rationalize the denominator of \frac{1}{2\sqrt{2}+\sqrt{7}} by multiplying numerator and denominator by 2\sqrt{2}-\sqrt{7}.

$$\frac{2\sqrt{2}-\sqrt{7}}{\left(2\sqrt{2}+\sqrt{7}\right)\left(2\sqrt{2}-\sqrt{7}\right)}$$

Consider \left(2\sqrt{2}+\sqrt{7}\right)\left(2\sqrt{2}-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{2}-\sqrt{7}}{\left(2\sqrt{2}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

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

$$\frac{2\sqrt{2}-\sqrt{7}}{2^{2}\left(\sqrt{2}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{2}-\sqrt{7}}{4\left(\sqrt{2}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}-\sqrt{7}}{4\times 2-\left(\sqrt{7}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{2\sqrt{2}-\sqrt{7}}{8-\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{2\sqrt{2}-\sqrt{7}}{8-7}$$

Subtract 7 from 8 to get 1.

$$\frac{2\sqrt{2}-\sqrt{7}}{1}$$

Anything divided by one gives itself.

$$2\sqrt{2}-\sqrt{7}$$