Solve frac{6-sqrt{3}}{2+sqrt{7}}

$$\frac { 6 - \sqrt { 3 } } { 2 + \sqrt { 7 } }$$

Evaluate

$$\frac{2\sqrt{3}}{3}-\frac{\sqrt{21}}{3}+2\sqrt{7}-4\approx 0.918677929$$

Solution Steps

Rationalize the denominator of \frac{6-\sqrt{3}}{2+\sqrt{7}} by multiplying numerator and denominator by 2-\sqrt{7}.

$$\frac{\left(6-\sqrt{3}\right)\left(2-\sqrt{7}\right)}{\left(2+\sqrt{7}\right)\left(2-\sqrt{7}\right)}$$

Consider \left(2+\sqrt{7}\right)\left(2-\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}.

$$\frac{\left(6-\sqrt{3}\right)\left(2-\sqrt{7}\right)}{2^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 2. Square \sqrt{7}.

$$\frac{\left(6-\sqrt{3}\right)\left(2-\sqrt{7}\right)}{4-7}$$

Subtract 7 from 4 to get -3.

$$\frac{\left(6-\sqrt{3}\right)\left(2-\sqrt{7}\right)}{-3}$$

Apply the distributive property by multiplying each term of 6-\sqrt{3} by each term of 2-\sqrt{7}.

$$\frac{12-6\sqrt{7}-2\sqrt{3}+\sqrt{3}\sqrt{7}}{-3}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{12-6\sqrt{7}-2\sqrt{3}+\sqrt{21}}{-3}$$

Multiply both numerator and denominator by -1.

$$\frac{-12+6\sqrt{7}+2\sqrt{3}-\sqrt{21}}{3}$$