Solve frac{sqrt{11}}{sqrt{7}-2}

$$\frac { \sqrt { 11 } } { \sqrt { 7 } - 2 }$$

Evaluate

$$\frac{\sqrt{11}\left(\sqrt{7}+2\right)}{3}\approx 5.136071323$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{7}-2} by multiplying numerator and denominator by \sqrt{7}+2.

$$\frac{\sqrt{11}\left(\sqrt{7}+2\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

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

$$\frac{\sqrt{11}\left(\sqrt{7}+2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}$$

Square \sqrt{7}. Square 2.

$$\frac{\sqrt{11}\left(\sqrt{7}+2\right)}{7-4}$$

Subtract 4 from 7 to get 3.

$$\frac{\sqrt{11}\left(\sqrt{7}+2\right)}{3}$$

Use the distributive property to multiply \sqrt{11} by \sqrt{7}+2.

$$\frac{\sqrt{11}\sqrt{7}+2\sqrt{11}}{3}$$

To multiply \sqrt{11} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{77}+2\sqrt{11}}{3}$$