Simplify 3 square root of 3+6 square root of 3-1/2 square root of 3

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

Combine $\sqrt{3}$ and $\frac{1}{2}$ .

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

Find the common denominator.

Write $3 \sqrt{3}$ as a fraction with denominator $1$ .

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

Multiply $\frac{3 \sqrt{3}}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{3 \sqrt{3}}{1}$ by $\frac{2}{2}$ .

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

Write $6 \sqrt{3}$ as a fraction with denominator $1$ .

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

Multiply $\frac{6 \sqrt{3}}{1}$ by $\frac{2}{2}$ .

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

Multiply $\frac{6 \sqrt{3}}{1}$ by $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $2$ by $3$ .

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

Multiply $2$ by $6$ .

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

Simplify by adding terms.

Add $6 \sqrt{3}$ and $12 \sqrt{3}$ .

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

Subtract $\sqrt{3}$ from $18 \sqrt{3}$ .

$\frac{17 \sqrt{3}}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{17 \sqrt{3}}{2}$

Decimal Form:

$14.72243186 \dots$