Solve frac{11-sqrt{7}}{sqrt{11}+sqrt{7}}*frac{sqrt{11}-sqrt{7}}{sqrt{11}-sqrt{7}}

$$\frac { 11 - \sqrt { 7 } } { \sqrt { 11 } + \sqrt { 7 } } \times \frac { \sqrt { 11 } - \sqrt { 7 } } { \sqrt { 11 } - \sqrt { 7 } }$$

Evaluate

$$\frac{11\sqrt{11}+7-\sqrt{77}-11\sqrt{7}}{4}\approx 1.401160971$$

Solution Steps

Divide \sqrt{11}-\sqrt{7} by \sqrt{11}-\sqrt{7} to get 1.

$$\frac{11-\sqrt{7}}{\sqrt{11}+\sqrt{7}}\times 1$$

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

$$\frac{\left(11-\sqrt{7}\right)\left(\sqrt{11}-\sqrt{7}\right)}{\left(\sqrt{11}+\sqrt{7}\right)\left(\sqrt{11}-\sqrt{7}\right)}\times 1$$

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}.

$$\frac{\left(11-\sqrt{7}\right)\left(\sqrt{11}-\sqrt{7}\right)}{\left(\sqrt{11}\right)^{2}-\left(\sqrt{7}\right)^{2}}\times 1$$

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

$$\frac{\left(11-\sqrt{7}\right)\left(\sqrt{11}-\sqrt{7}\right)}{11-7}\times 1$$

Subtract 7 from 11 to get 4.

$$\frac{\left(11-\sqrt{7}\right)\left(\sqrt{11}-\sqrt{7}\right)}{4}\times 1$$

Express \frac{\left(11-\sqrt{7}\right)\left(\sqrt{11}-\sqrt{7}\right)}{4}\times 1 as a single fraction.

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

Apply the distributive property by multiplying each term of 11-\sqrt{7} by each term of \sqrt{11}-\sqrt{7}.

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

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

$$\frac{11\sqrt{11}-11\sqrt{7}-\sqrt{77}+\left(\sqrt{7}\right)^{2}}{4}$$

The square of \sqrt{7} is 7.

$$\frac{11\sqrt{11}-11\sqrt{7}-\sqrt{77}+7}{4}$$