Solve frac{1/sqrt{2}-1/sqrt{3}}{1-1/sqrt{6}}

$$\frac { \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 } } } { 1 - \frac { 1 } { \sqrt { 6 } } }$$

Evaluate

$$\frac{2\sqrt{2}-\sqrt{3}}{5}\approx 0.219275263$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

$$\frac{\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{6}}}$$

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

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

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{3}}{1-\frac{1}{\sqrt{6}}}$$

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}.

$$\frac{\frac{3\sqrt{2}}{6}-\frac{2\sqrt{3}}{6}}{1-\frac{1}{\sqrt{6}}}$$

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

$$\frac{\frac{3\sqrt{2}-2\sqrt{3}}{6}}{1-\frac{1}{\sqrt{6}}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

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

The square of \sqrt{6} is 6.

$$\frac{\frac{3\sqrt{2}-2\sqrt{3}}{6}}{1-\frac{\sqrt{6}}{6}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{6}{6}.

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

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

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

Divide \frac{3\sqrt{2}-2\sqrt{3}}{6} by \frac{6-\sqrt{6}}{6} by multiplying \frac{3\sqrt{2}-2\sqrt{3}}{6} by the reciprocal of \frac{6-\sqrt{6}}{6}.

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

Cancel out 6 in both numerator and denominator.

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

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

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

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

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

Expand \left(-\sqrt{6}\right)^{2}.

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

Calculate -1 to the power of 2 and get 1.

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

The square of \sqrt{6} is 6.

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

Multiply 1 and 6 to get 6.

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

Calculate 6 to the power of 2 and get 36.

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

Subtract 36 from 6 to get -30.

$$\frac{\left(-2\sqrt{3}+3\sqrt{2}\right)\left(-\sqrt{6}-6\right)}{-30}$$

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

$$\frac{2\sqrt{3}\sqrt{6}+12\sqrt{3}-3\sqrt{2}\sqrt{6}-18\sqrt{2}}{-30}$$

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}.

$$\frac{2\sqrt{3}\sqrt{3}\sqrt{2}+12\sqrt{3}-3\sqrt{2}\sqrt{6}-18\sqrt{2}}{-30}$$

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

$$\frac{2\times 3\sqrt{2}+12\sqrt{3}-3\sqrt{2}\sqrt{6}-18\sqrt{2}}{-30}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{2}+12\sqrt{3}-3\sqrt{2}\sqrt{6}-18\sqrt{2}}{-30}$$

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}.

$$\frac{6\sqrt{2}+12\sqrt{3}-3\sqrt{2}\sqrt{2}\sqrt{3}-18\sqrt{2}}{-30}$$

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

$$\frac{6\sqrt{2}+12\sqrt{3}-3\times 2\sqrt{3}-18\sqrt{2}}{-30}$$

Multiply -3 and 2 to get -6.

$$\frac{6\sqrt{2}+12\sqrt{3}-6\sqrt{3}-18\sqrt{2}}{-30}$$

Combine 12\sqrt{3} and -6\sqrt{3} to get 6\sqrt{3}.

$$\frac{6\sqrt{2}+6\sqrt{3}-18\sqrt{2}}{-30}$$

Combine 6\sqrt{2} and -18\sqrt{2} to get -12\sqrt{2}.

$$\frac{-12\sqrt{2}+6\sqrt{3}}{-30}$$