Solve frac{3sqrt{3}+1}{sqrt{2}+sqrt{7}}

$$\frac { 3 \sqrt { 3 } + 1 } { \sqrt { 2 } + \sqrt { 7 } }$$

Evaluate

$$\frac{\sqrt{7}+3\sqrt{21}-\sqrt{2}-3\sqrt{6}}{5}\approx 1.526159121$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{3}+1}{\sqrt{2}+\sqrt{7}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{7}.

$$\frac{\left(3\sqrt{3}+1\right)\left(\sqrt{2}-\sqrt{7}\right)}{\left(\sqrt{2}+\sqrt{7}\right)\left(\sqrt{2}-\sqrt{7}\right)}$$

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

$$\frac{\left(3\sqrt{3}+1\right)\left(\sqrt{2}-\sqrt{7}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{2}. Square \sqrt{7}.

$$\frac{\left(3\sqrt{3}+1\right)\left(\sqrt{2}-\sqrt{7}\right)}{2-7}$$

Subtract 7 from 2 to get -5.

$$\frac{\left(3\sqrt{3}+1\right)\left(\sqrt{2}-\sqrt{7}\right)}{-5}$$

Apply the distributive property by multiplying each term of 3\sqrt{3}+1 by each term of \sqrt{2}-\sqrt{7}.

$$\frac{3\sqrt{3}\sqrt{2}-3\sqrt{3}\sqrt{7}+\sqrt{2}-\sqrt{7}}{-5}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{6}-3\sqrt{3}\sqrt{7}+\sqrt{2}-\sqrt{7}}{-5}$$

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

$$\frac{3\sqrt{6}-3\sqrt{21}+\sqrt{2}-\sqrt{7}}{-5}$$

Multiply both numerator and denominator by -1.

$$\frac{-3\sqrt{6}+3\sqrt{21}-\sqrt{2}+\sqrt{7}}{5}$$