Solve 27/sqrt{11+sqrt{2}}

$$\frac { 27 } { \sqrt { 11 } + \sqrt { 2 } }$$

Evaluate

$$3\left(\sqrt{11}-\sqrt{2}\right)\approx 5.707233684$$

Solution Steps

Rationalize the denominator of \frac{27}{\sqrt{11}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{11}-\sqrt{2}.

$$\frac{27\left(\sqrt{11}-\sqrt{2}\right)}{\left(\sqrt{11}+\sqrt{2}\right)\left(\sqrt{11}-\sqrt{2}\right)}$$

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

$$\frac{27\left(\sqrt{11}-\sqrt{2}\right)}{\left(\sqrt{11}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{11}. Square \sqrt{2}.

$$\frac{27\left(\sqrt{11}-\sqrt{2}\right)}{11-2}$$

Subtract 2 from 11 to get 9.

$$\frac{27\left(\sqrt{11}-\sqrt{2}\right)}{9}$$

Divide 27\left(\sqrt{11}-\sqrt{2}\right) by 9 to get 3\left(\sqrt{11}-\sqrt{2}\right).

$$3\left(\sqrt{11}-\sqrt{2}\right)$$

Use the distributive property to multiply 3 by \sqrt{11}-\sqrt{2}.

$$3\sqrt{11}-3\sqrt{2}$$