Solve 1/sqrt{5-2sqrt{6}}

$$\frac { 1 } { \sqrt { 5 } - 2 \sqrt { 6 } }$$

Evaluate

$$\frac{-\sqrt{5}-2\sqrt{6}}{19}\approx -0.375528814$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{5}-2\sqrt{6}} by multiplying numerator and denominator by \sqrt{5}+2\sqrt{6}.

$$\frac{\sqrt{5}+2\sqrt{6}}{\left(\sqrt{5}-2\sqrt{6}\right)\left(\sqrt{5}+2\sqrt{6}\right)}$$

Consider \left(\sqrt{5}-2\sqrt{6}\right)\left(\sqrt{5}+2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{5}+2\sqrt{6}}{\left(\sqrt{5}\right)^{2}-\left(-2\sqrt{6}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{5}+2\sqrt{6}}{5-\left(-2\sqrt{6}\right)^{2}}$$

Expand \left(-2\sqrt{6}\right)^{2}.

$$\frac{\sqrt{5}+2\sqrt{6}}{5-\left(-2\right)^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\sqrt{5}+2\sqrt{6}}{5-4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{5}+2\sqrt{6}}{5-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{\sqrt{5}+2\sqrt{6}}{5-24}$$

Subtract 24 from 5 to get -19.

$$\frac{\sqrt{5}+2\sqrt{6}}{-19}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{5}-2\sqrt{6}}{19}$$