Solve (11/6-sqrt{3})(frac{6+sqrt{3}}{6+sqrt{3}})

$$( \frac { 11 } { 6 - \sqrt { 3 } } ) ( \frac { 6 + \sqrt { 3 } } { 6 + \sqrt { 3 } } )$$

Evaluate

$$\frac{\sqrt{3}}{3}+2\approx 2.577350269$$

Solution Steps

Divide 6+\sqrt{3} by 6+\sqrt{3} to get 1.

$$\frac{11}{6-\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{11}{6-\sqrt{3}} by multiplying numerator and denominator by 6+\sqrt{3}.

$$\frac{11\left(6+\sqrt{3}\right)}{\left(6-\sqrt{3}\right)\left(6+\sqrt{3}\right)}\times 1$$

Consider \left(6-\sqrt{3}\right)\left(6+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{11\left(6+\sqrt{3}\right)}{6^{2}-\left(\sqrt{3}\right)^{2}}\times 1$$

Square 6. Square \sqrt{3}.

$$\frac{11\left(6+\sqrt{3}\right)}{36-3}\times 1$$

Subtract 3 from 36 to get 33.

$$\frac{11\left(6+\sqrt{3}\right)}{33}\times 1$$

Divide 11\left(6+\sqrt{3}\right) by 33 to get \frac{1}{3}\left(6+\sqrt{3}\right).

$$\frac{1}{3}\left(6+\sqrt{3}\right)\times 1$$

Use the distributive property to multiply \frac{1}{3} by 6+\sqrt{3}.

$$\left(\frac{1}{3}\times 6+\frac{1}{3}\sqrt{3}\right)\times 1$$

Multiply \frac{1}{3} and 6 to get \frac{6}{3}.

$$\left(\frac{6}{3}+\frac{1}{3}\sqrt{3}\right)\times 1$$

Divide 6 by 3 to get 2.

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

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

$$2+\frac{1}{3}\sqrt{3}$$