Solve frac{3sqrt{2}}{sqrt{3}+sqrt{6}}-frac{4sqrt{3}}{sqrt{2}+sqrt{6}}=

$$\frac { 3 \sqrt { 2 } } { \sqrt { 3 } + \sqrt { 6 } } - \frac { 4 \sqrt { 3 } } { \sqrt { 2 } + \sqrt { 6 } } =$$

Evaluate

$$\sqrt{2}\left(\sqrt{6}-3\right)\approx -0.778539072$$

Solution Steps

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

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

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

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

Square \sqrt{3}. Square \sqrt{6}.

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

Subtract 6 from 3 to get -3.

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

Rationalize the denominator of \frac{4\sqrt{3}}{\sqrt{2}+\sqrt{6}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{6}.

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

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

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

Square \sqrt{2}. Square \sqrt{6}.

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

Subtract 6 from 2 to get -4.

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

Cancel out -3 and -3.

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

Use the distributive property to multiply -\sqrt{2} by \sqrt{3}-\sqrt{6}.

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

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

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

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}.

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

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

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

Cancel out -4 and -4.

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

Use the distributive property to multiply -\sqrt{3} by \sqrt{2}-\sqrt{6}.

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

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

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

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

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

To find the opposite of -\sqrt{6}+3\sqrt{2}, find the opposite of each term.

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

The opposite of -\sqrt{6} is \sqrt{6}.

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

Combine -\sqrt{6} and \sqrt{6} to get 0.

$$2\sqrt{3}-3\sqrt{2}$$

Factor

$$\sqrt{2} {(\sqrt{2} \sqrt{3} - 3)} = -0.778539072$$