Solve 2(1/sqrt{2}+sqrt{3})-6(1/sqrt{2}-1/sqrt{3})*6

$$2 ( \frac { 1 } { \sqrt { 2 } } + \sqrt { 3 } ) - 6 ( \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 } } ) \cdot 6$$

Evaluate

$$14\sqrt{3}-17\sqrt{2}\approx 0.207080746$$

Short Solution Steps

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

$$2\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{3}\right)-6\times 6\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)$$

The square of \sqrt{2} is 2.

$$2\left(\frac{\sqrt{2}}{2}+\sqrt{3}\right)-6\times 6\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3} times \frac{2}{2}.

$$2\left(\frac{\sqrt{2}}{2}+\frac{2\sqrt{3}}{2}\right)-6\times 6\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)$$

Since \frac{\sqrt{2}}{2} and \frac{2\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

$$2\times \frac{\sqrt{2}+2\sqrt{3}}{2}-6\times 6\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)$$

Cancel out 2 and 2.

$$\sqrt{2}+2\sqrt{3}-6\times 6\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)$$

Multiply 6 and 6 to get 36.

$$\sqrt{2}+2\sqrt{3}-36\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)$$

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

$$\sqrt{2}+2\sqrt{3}-36\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{1}{\sqrt{3}}\right)$$

The square of \sqrt{2} is 2.

$$\sqrt{2}+2\sqrt{3}-36\left(\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{3}}\right)$$

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

$$\sqrt{2}+2\sqrt{3}-36\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)$$

The square of \sqrt{3} is 3.

$$\sqrt{2}+2\sqrt{3}-36\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{\sqrt{2}}{2} times \frac{3}{3}. Multiply \frac{\sqrt{3}}{3} times \frac{2}{2}.

$$\sqrt{2}+2\sqrt{3}-36\left(\frac{3\sqrt{2}}{6}-\frac{2\sqrt{3}}{6}\right)$$

Since \frac{3\sqrt{2}}{6} and \frac{2\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{2}+2\sqrt{3}-36\times \frac{3\sqrt{2}-2\sqrt{3}}{6}$$

Cancel out 6, the greatest common factor in 36 and 6.

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

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

$$\sqrt{2}+2\sqrt{3}-18\sqrt{2}+12\sqrt{3}$$

Combine \sqrt{2} and -18\sqrt{2} to get -17\sqrt{2}.

$$-17\sqrt{2}+2\sqrt{3}+12\sqrt{3}$$

Combine 2\sqrt{3} and 12\sqrt{3} to get 14\sqrt{3}.

$$-17\sqrt{2}+14\sqrt{3}$$

Factor

$$14 \sqrt{3} - 17 \sqrt{2} = 0.207080746$$