Solve 7/sqrt{11-2}+5/4+sqrt{11}

$$\frac { 7 } { \sqrt { 11 } - 2 } + \frac { 5 } { 4 + \sqrt { 11 } }$$

Evaluate

$$6$$

Solution Steps

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

$$\frac{7\left(\sqrt{11}+2\right)}{\left(\sqrt{11}-2\right)\left(\sqrt{11}+2\right)}+\frac{5}{4+\sqrt{11}}$$

Consider \left(\sqrt{11}-2\right)\left(\sqrt{11}+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}.

$$\frac{7\left(\sqrt{11}+2\right)}{\left(\sqrt{11}\right)^{2}-2^{2}}+\frac{5}{4+\sqrt{11}}$$

Square \sqrt{11}. Square 2.

$$\frac{7\left(\sqrt{11}+2\right)}{11-4}+\frac{5}{4+\sqrt{11}}$$

Subtract 4 from 11 to get 7.

$$\frac{7\left(\sqrt{11}+2\right)}{7}+\frac{5}{4+\sqrt{11}}$$

Cancel out 7 and 7.

$$\sqrt{11}+2+\frac{5}{4+\sqrt{11}}$$

Rationalize the denominator of \frac{5}{4+\sqrt{11}} by multiplying numerator and denominator by 4-\sqrt{11}.

$$\sqrt{11}+2+\frac{5\left(4-\sqrt{11}\right)}{\left(4+\sqrt{11}\right)\left(4-\sqrt{11}\right)}$$

Consider \left(4+\sqrt{11}\right)\left(4-\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\sqrt{11}+2+\frac{5\left(4-\sqrt{11}\right)}{4^{2}-\left(\sqrt{11}\right)^{2}}$$

Square 4. Square \sqrt{11}.

$$\sqrt{11}+2+\frac{5\left(4-\sqrt{11}\right)}{16-11}$$

Subtract 11 from 16 to get 5.

$$\sqrt{11}+2+\frac{5\left(4-\sqrt{11}\right)}{5}$$

Cancel out 5 and 5.

$$\sqrt{11}+2+4-\sqrt{11}$$

Add 2 and 4 to get 6.

$$\sqrt{11}+6-\sqrt{11}$$

Combine \sqrt{11} and -\sqrt{11} to get 0.

$$6$$

Factor

$$2\times 3$$