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

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

Evaluate

$$\frac{6\sqrt{3}+9\sqrt{42}-\sqrt{14}-21}{61}\approx 0.72093957$$

Solution Steps

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

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

Consider \left(\sqrt{2}-3\sqrt{7}\right)\left(\sqrt{2}+3\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}.

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

The square of \sqrt{2} is 2.

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

Expand \left(-3\sqrt{7}\right)^{2}.

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

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{2-9\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{2-9\times 7}$$

Multiply 9 and 7 to get 63.

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{2-63}$$

Subtract 63 from 2 to get -61.

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{-61}$$

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

$$\frac{\sqrt{7}\sqrt{2}+3\left(\sqrt{7}\right)^{2}-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

To multiply \sqrt{7} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{14}+3\left(\sqrt{7}\right)^{2}-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{14}+3\times 7-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

Multiply 3 and 7 to get 21.

$$\frac{\sqrt{14}+21-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{\sqrt{14}+21-3\sqrt{2}\sqrt{3}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{\sqrt{14}+21-3\times 2\sqrt{3}-9\sqrt{6}\sqrt{7}}{-61}$$

Multiply -3 and 2 to get -6.

$$\frac{\sqrt{14}+21-6\sqrt{3}-9\sqrt{6}\sqrt{7}}{-61}$$

To multiply \sqrt{6} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{14}+21-6\sqrt{3}-9\sqrt{42}}{-61}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{14}-21+6\sqrt{3}+9\sqrt{42}}{61}$$