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

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

Evaluate

$$-\sqrt{5}\approx -2.236067977$$

Solution Steps

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

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

Consider \left(\sqrt{5}-\sqrt{6}\right)\left(\sqrt{5}+\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}.

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

Square \sqrt{5}. Square \sqrt{6}.

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

Subtract 6 from 5 to get -1.

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

Anything divided by -1 gives its opposite. To find the opposite of \sqrt{5}+\sqrt{6}, find the opposite of each term.

$$-\sqrt{5}-\sqrt{6}+\sqrt{6}$$

Combine -\sqrt{6} and \sqrt{6} to get 0.

$$-\sqrt{5}$$