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

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

Evaluate

$$6\left(\sqrt{3}-\sqrt{2}\right)\approx 1.907023471$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

The square of \sqrt{3} is 3.

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

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}.

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

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

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

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

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

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

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

$$\frac{\sqrt{6}\times 6\left(3\sqrt{2}-2\sqrt{3}\right)}{18-\left(2\sqrt{3}\right)^{2}}$$

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

$$\frac{\sqrt{6}\times 6\left(3\sqrt{2}-2\sqrt{3}\right)}{18-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\sqrt{6}\times 6\left(3\sqrt{2}-2\sqrt{3}\right)}{18-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{6}\times 6\left(3\sqrt{2}-2\sqrt{3}\right)}{18-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\sqrt{6}\times 6\left(3\sqrt{2}-2\sqrt{3}\right)}{18-12}$$

Subtract 12 from 18 to get 6.

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

Cancel out 6 and 6.

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

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

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

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}.

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

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

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

Multiply 3 and 2 to get 6.

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

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}.

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

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

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

Multiply -2 and 3 to get -6.

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