Solve 4/sqrt{7+sqrt{11}}+sqrt{99}-sqrt{63}

$$\frac { 4 } { \sqrt { 7 } + \sqrt { 11 } } + \sqrt { 99 } - \sqrt { 63 }$$

Evaluate

$$4\left(\sqrt{11}-\sqrt{7}\right)\approx 2.683493917$$

Solution Steps

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

$$\frac{4\left(\sqrt{7}-\sqrt{11}\right)}{\left(\sqrt{7}+\sqrt{11}\right)\left(\sqrt{7}-\sqrt{11}\right)}+\sqrt{99}-\sqrt{63}$$

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

$$\frac{4\left(\sqrt{7}-\sqrt{11}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{11}\right)^{2}}+\sqrt{99}-\sqrt{63}$$

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

$$\frac{4\left(\sqrt{7}-\sqrt{11}\right)}{7-11}+\sqrt{99}-\sqrt{63}$$

Subtract 11 from 7 to get -4.

$$\frac{4\left(\sqrt{7}-\sqrt{11}\right)}{-4}+\sqrt{99}-\sqrt{63}$$

Cancel out -4 and -4.

$$-\left(\sqrt{7}-\sqrt{11}\right)+\sqrt{99}-\sqrt{63}$$

Factor 99=3^{2}\times 11. Rewrite the square root of the product \sqrt{3^{2}\times 11} as the product of square roots \sqrt{3^{2}}\sqrt{11}. Take the square root of 3^{2}.

$$-\left(\sqrt{7}-\sqrt{11}\right)+3\sqrt{11}-\sqrt{63}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

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

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

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

The opposite of -\sqrt{11} is \sqrt{11}.

$$-\sqrt{7}+\sqrt{11}+3\sqrt{11}-3\sqrt{7}$$

Combine \sqrt{11} and 3\sqrt{11} to get 4\sqrt{11}.

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

Combine -\sqrt{7} and -3\sqrt{7} to get -4\sqrt{7}.

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