Solve 1+sqrt{3}-3*1sqrt{3}sqrt{3/2}

$$1+ \sqrt{ 3 } -3 \cdot 1 \sqrt{ 3 } \sqrt{ \frac{ 3 }{ 2 } }$$

Evaluate

$$\frac{2\sqrt{3}}{2}-\frac{9\sqrt{2}}{2}+1\approx -3.631910223$$

Short Solution Steps

Multiply 1 and 3 to get 3.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

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

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

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}.

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

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

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

Multiply 3 and 3 to get 9.

$$1+\sqrt{3}-\frac{9\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1+\sqrt{3} times \frac{2}{2}.

$$\frac{2\left(1+\sqrt{3}\right)}{2}-\frac{9\sqrt{2}}{2}$$

Since \frac{2\left(1+\sqrt{3}\right)}{2} and \frac{9\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(1+\sqrt{3}\right)-9\sqrt{2}}{2}$$

Do the multiplications in 2\left(1+\sqrt{3}\right)-9\sqrt{2}.

$$\frac{2+2\sqrt{3}-9\sqrt{2}}{2}$$