Solve frac{2(sqrt{3}+sqrt{2})}{4-sqrt{6}}*frac{9-sqrt{6}}{4-sqrt{6}}

$$\frac { 2 ( \sqrt { 3 } + \sqrt { 2 } ) } { 4 - \sqrt { 6 } } \cdot \frac { 9 - \sqrt { 6 } } { 4 - \sqrt { 6 } }$$

Evaluate

$$5\sqrt{3}+6\sqrt{2}\approx 17.145535412$$

Solution Steps

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

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{\left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right)}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Consider \left(4-\sqrt{6}\right)\left(4+\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}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{4^{2}-\left(\sqrt{6}\right)^{2}}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Square 4. Square \sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{16-6}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Subtract 6 from 16 to get 10.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Rationalize the denominator of \frac{9-\sqrt{6}}{4-\sqrt{6}} by multiplying numerator and denominator by 4+\sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{\left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right)}$$

Consider \left(4-\sqrt{6}\right)\left(4+\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}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{4^{2}-\left(\sqrt{6}\right)^{2}}$$

Square 4. Square \sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{16-6}$$

Subtract 6 from 16 to get 10.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{10}$$

Multiply \frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10} times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{10\times 10}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(\sqrt{6}+4\right)\left(\sqrt{6}+4\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Multiply \sqrt{6}+4 and \sqrt{6}+4 to get \left(\sqrt{6}+4\right)^{2}.

$$\frac{\left(\sqrt{6}+4\right)^{2}\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{6}+4\right)^{2}.

$$\frac{\left(\left(\sqrt{6}\right)^{2}+8\sqrt{6}+16\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

The square of \sqrt{6} is 6.

$$\frac{\left(6+8\sqrt{6}+16\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Add 6 and 16 to get 22.

$$\frac{\left(22+8\sqrt{6}\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Multiply 5 and 10 to get 50.

$$\frac{\left(22+8\sqrt{6}\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Apply the distributive property by multiplying each term of 22+8\sqrt{6} by each term of -\sqrt{6}+9.

$$\frac{\left(-22\sqrt{6}+198-8\left(\sqrt{6}\right)^{2}+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

The square of \sqrt{6} is 6.

$$\frac{\left(-22\sqrt{6}+198-8\times 6+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Multiply -8 and 6 to get -48.

$$\frac{\left(-22\sqrt{6}+198-48+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Subtract 48 from 198 to get 150.

$$\frac{\left(-22\sqrt{6}+150+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Combine -22\sqrt{6} and 72\sqrt{6} to get 50\sqrt{6}.

$$\frac{\left(50\sqrt{6}+150\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Apply the distributive property by multiplying each term of 50\sqrt{6}+150 by each term of \sqrt{2}+\sqrt{3}.

$$\frac{50\sqrt{6}\sqrt{2}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

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}.

$$\frac{50\sqrt{2}\sqrt{3}\sqrt{2}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

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

$$\frac{50\times 2\sqrt{3}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

Multiply 50 and 2 to get 100.

$$\frac{100\sqrt{3}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

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}.

$$\frac{100\sqrt{3}+50\sqrt{3}\sqrt{2}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

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

$$\frac{100\sqrt{3}+50\times 3\sqrt{2}+150\sqrt{2}+150\sqrt{3}}{50}$$

Multiply 50 and 3 to get 150.

$$\frac{100\sqrt{3}+150\sqrt{2}+150\sqrt{2}+150\sqrt{3}}{50}$$

Combine 150\sqrt{2} and 150\sqrt{2} to get 300\sqrt{2}.

$$\frac{100\sqrt{3}+300\sqrt{2}+150\sqrt{3}}{50}$$

Combine 100\sqrt{3} and 150\sqrt{3} to get 250\sqrt{3}.

$$\frac{250\sqrt{3}+300\sqrt{2}}{50}$$

Divide each term of 250\sqrt{3}+300\sqrt{2} by 50 to get 5\sqrt{3}+6\sqrt{2}.

$$5\sqrt{3}+6\sqrt{2}$$

Factor

$$5 \sqrt{3} + 6 \sqrt{2} = 17.145535412$$