Solve 5/sqrt{6+sqrt{7}}

$$\frac { 5 } { \sqrt { 6 } + \sqrt { 7 } }$$

Evaluate

$$5\left(\sqrt{7}-\sqrt{6}\right)\approx 0.981307841$$

Solution Steps

Rationalize the denominator of \frac{5}{\sqrt{6}+\sqrt{7}} by multiplying numerator and denominator by \sqrt{6}-\sqrt{7}.

$$\frac{5\left(\sqrt{6}-\sqrt{7}\right)}{\left(\sqrt{6}+\sqrt{7}\right)\left(\sqrt{6}-\sqrt{7}\right)}$$

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

$$\frac{5\left(\sqrt{6}-\sqrt{7}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

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

$$\frac{5\left(\sqrt{6}-\sqrt{7}\right)}{6-7}$$

Subtract 7 from 6 to get -1.

$$\frac{5\left(\sqrt{6}-\sqrt{7}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-5\left(\sqrt{6}-\sqrt{7}\right)$$

Use the distributive property to multiply -5 by \sqrt{6}-\sqrt{7}.

$$-5\sqrt{6}+5\sqrt{7}$$