Solve -3/sqrt{2+sqrt{11}}

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

Evaluate

$$\frac{\sqrt{2}-\sqrt{11}}{3}\approx -0.634137076$$

Solution Steps

Rationalize the denominator of \frac{-3}{\sqrt{2}+\sqrt{11}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{11}.

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

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

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

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

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

Subtract 11 from 2 to get -9.

$$\frac{-3\left(\sqrt{2}-\sqrt{11}\right)}{-9}$$

Divide -3\left(\sqrt{2}-\sqrt{11}\right) by -9 to get \frac{1}{3}\left(\sqrt{2}-\sqrt{11}\right).

$$\frac{1}{3}\left(\sqrt{2}-\sqrt{11}\right)$$

Use the distributive property to multiply \frac{1}{3} by \sqrt{2}-\sqrt{11}.

$$\frac{1}{3}\sqrt{2}+\frac{1}{3}\left(-1\right)\sqrt{11}$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

$$\frac{1}{3}\sqrt{2}-\frac{1}{3}\sqrt{11}$$