Solve 1/sqrt{7-sqrt{6}}

$$\frac { 1 } { \sqrt { 7 } - \sqrt { 6 } }$$

Evaluate

$$\sqrt{6}+\sqrt{7}\approx 5.095241054$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{7}-\sqrt{6}} by multiplying numerator and denominator by \sqrt{7}+\sqrt{6}.

$$\frac{\sqrt{7}+\sqrt{6}}{\left(\sqrt{7}-\sqrt{6}\right)\left(\sqrt{7}+\sqrt{6}\right)}$$

Consider \left(\sqrt{7}-\sqrt{6}\right)\left(\sqrt{7}+\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}.

$$\frac{\sqrt{7}+\sqrt{6}}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Square \sqrt{7}. Square \sqrt{6}.

$$\frac{\sqrt{7}+\sqrt{6}}{7-6}$$

Subtract 6 from 7 to get 1.

$$\frac{\sqrt{7}+\sqrt{6}}{1}$$

Anything divided by one gives itself.

$$\sqrt{7}+\sqrt{6}$$