Solve 2/3*(1/sqrt{10})-1/2*(3/sqrt{10})=

$$\frac{ 2 }{ 3 } \times ( \frac{ 1 }{ \sqrt{ 10 } } )- \frac{ 1 }{ 2 } \times ( \frac{ 3 }{ \sqrt{ 10 } } )=$$

Evaluate

$$-\frac{\sqrt{10}}{12}\approx -0.263523138$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{2}{3}\times \frac{\sqrt{10}}{\left(\sqrt{10}\right)^{2}}-\frac{1}{2}\times \frac{3}{\sqrt{10}}$$

The square of \sqrt{10} is 10.

$$\frac{2}{3}\times \frac{\sqrt{10}}{10}-\frac{1}{2}\times \frac{3}{\sqrt{10}}$$

Multiply \frac{2}{3} times \frac{\sqrt{10}}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{10}}{3\times 10}-\frac{1}{2}\times \frac{3}{\sqrt{10}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{10}}{3\times 5}-\frac{1}{2}\times \frac{3}{\sqrt{10}}$$

Rationalize the denominator of \frac{3}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{10}}{3\times 5}-\frac{1}{2}\times \frac{3\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{10}}{3\times 5}-\frac{1}{2}\times \frac{3\sqrt{10}}{10}$$

Multiply \frac{1}{2} times \frac{3\sqrt{10}}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{10}}{3\times 5}-\frac{3\sqrt{10}}{2\times 10}$$

Multiply 2 and 10 to get 20.

$$\frac{\sqrt{10}}{3\times 5}-\frac{3\sqrt{10}}{20}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\times 5 and 20 is 60. Multiply \frac{\sqrt{10}}{3\times 5} times \frac{4}{4}. Multiply \frac{3\sqrt{10}}{20} times \frac{3}{3}.

$$\frac{4\sqrt{10}}{60}-\frac{3\times 3\sqrt{10}}{60}$$

Since \frac{4\sqrt{10}}{60} and \frac{3\times 3\sqrt{10}}{60} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\sqrt{10}-3\times 3\sqrt{10}}{60}$$

Do the multiplications in 4\sqrt{10}-3\times 3\sqrt{10}.

$$\frac{4\sqrt{10}-9\sqrt{10}}{60}$$

Do the calculations in 4\sqrt{10}-9\sqrt{10}.

$$\frac{-5\sqrt{10}}{60}$$

Divide -5\sqrt{10} by 60 to get -\frac{1}{12}\sqrt{10}.

$$-\frac{1}{12}\sqrt{10}$$