Solve 5/4-sqrt{11}-4/sqrt{11-sqrt{7}}-2/3+sqrt{7}

$$\frac { 5 } { 4 - \sqrt { 11 } } - \frac { 4 } { \sqrt { 11 } - \sqrt { 7 } } - \frac { 2 } { 3 + \sqrt { 7 } }$$

Evaluate

$$1$$

Solution Steps

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

$$\frac{5\left(4+\sqrt{11}\right)}{\left(4-\sqrt{11}\right)\left(4+\sqrt{11}\right)}-\frac{4}{\sqrt{11}-\sqrt{7}}-\frac{2}{3+\sqrt{7}}$$

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}.

$$\frac{5\left(4+\sqrt{11}\right)}{4^{2}-\left(\sqrt{11}\right)^{2}}-\frac{4}{\sqrt{11}-\sqrt{7}}-\frac{2}{3+\sqrt{7}}$$

Square 4. Square \sqrt{11}.

$$\frac{5\left(4+\sqrt{11}\right)}{16-11}-\frac{4}{\sqrt{11}-\sqrt{7}}-\frac{2}{3+\sqrt{7}}$$

Subtract 11 from 16 to get 5.

$$\frac{5\left(4+\sqrt{11}\right)}{5}-\frac{4}{\sqrt{11}-\sqrt{7}}-\frac{2}{3+\sqrt{7}}$$

Cancel out 5 and 5.

$$4+\sqrt{11}-\frac{4}{\sqrt{11}-\sqrt{7}}-\frac{2}{3+\sqrt{7}}$$

Rationalize the denominator of \frac{4}{\sqrt{11}-\sqrt{7}} by multiplying numerator and denominator by \sqrt{11}+\sqrt{7}.

$$4+\sqrt{11}-\frac{4\left(\sqrt{11}+\sqrt{7}\right)}{\left(\sqrt{11}-\sqrt{7}\right)\left(\sqrt{11}+\sqrt{7}\right)}-\frac{2}{3+\sqrt{7}}$$

Consider \left(\sqrt{11}-\sqrt{7}\right)\left(\sqrt{11}+\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}.

$$4+\sqrt{11}-\frac{4\left(\sqrt{11}+\sqrt{7}\right)}{\left(\sqrt{11}\right)^{2}-\left(\sqrt{7}\right)^{2}}-\frac{2}{3+\sqrt{7}}$$

Square \sqrt{11}. Square \sqrt{7}.

$$4+\sqrt{11}-\frac{4\left(\sqrt{11}+\sqrt{7}\right)}{11-7}-\frac{2}{3+\sqrt{7}}$$

Subtract 7 from 11 to get 4.

$$4+\sqrt{11}-\frac{4\left(\sqrt{11}+\sqrt{7}\right)}{4}-\frac{2}{3+\sqrt{7}}$$

Cancel out 4 and 4.

$$4+\sqrt{11}-\left(\sqrt{11}+\sqrt{7}\right)-\frac{2}{3+\sqrt{7}}$$

To find the opposite of \sqrt{11}+\sqrt{7}, find the opposite of each term.

$$4+\sqrt{11}-\sqrt{11}-\sqrt{7}-\frac{2}{3+\sqrt{7}}$$

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

$$4-\sqrt{7}-\frac{2}{3+\sqrt{7}}$$

Rationalize the denominator of \frac{2}{3+\sqrt{7}} by multiplying numerator and denominator by 3-\sqrt{7}.

$$4-\sqrt{7}-\frac{2\left(3-\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}$$

Consider \left(3+\sqrt{7}\right)\left(3-\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}.

$$4-\sqrt{7}-\frac{2\left(3-\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 3. Square \sqrt{7}.

$$4-\sqrt{7}-\frac{2\left(3-\sqrt{7}\right)}{9-7}$$

Subtract 7 from 9 to get 2.

$$4-\sqrt{7}-\frac{2\left(3-\sqrt{7}\right)}{2}$$

Cancel out 2 and 2.

$$4-\sqrt{7}-\left(3-\sqrt{7}\right)$$

To find the opposite of 3-\sqrt{7}, find the opposite of each term.

$$4-\sqrt{7}-3-\left(-\sqrt{7}\right)$$

The opposite of -\sqrt{7} is \sqrt{7}.

$$4-\sqrt{7}-3+\sqrt{7}$$

Subtract 3 from 4 to get 1.

$$1-\sqrt{7}+\sqrt{7}$$

Combine -\sqrt{7} and \sqrt{7} to get 0.

$$1$$

Factor

$$1$$