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

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

Evaluate

$$-\frac{3\sqrt{3}}{2}+6-\sqrt{6}\approx 0.952434046$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

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

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

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

The square of \sqrt{2} is 2.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

Cancel out 2 and 2.

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

Express 3\times \frac{\sqrt{3}}{2} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 6-\sqrt{6} times \frac{2}{2}.

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

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

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

Do the multiplications in 2\left(6-\sqrt{6}\right)-3\sqrt{3}.

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

Factor

$$\frac{12 - 2 \sqrt{6} - 3 \sqrt{3}}{2} = 0.9524340458635061$$