Solve -2-sqrt{3}+sqrt{3/2}*sqrt{18}

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

Evaluate

$$2\left(\sqrt{3}-1\right)\approx 1.464101615$$

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}}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

Express \frac{\sqrt{6}}{2}\times 3 as a single fraction.

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

Express \frac{\sqrt{6}\times 3}{2}\sqrt{2} as a single fraction.

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

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

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

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

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

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

$$\frac{-4-2\sqrt{3}+6\sqrt{3}}{2}$$

Do the calculations in -4-2\sqrt{3}+6\sqrt{3}.

$$\frac{-4+4\sqrt{3}}{2}$$