Use this rule: \(a \times \frac{b}{c}=\frac{ab}{c}\).
\[\frac{1+\frac{\sqrt{3}\sqrt{3}}{3}}{\sqrt{3}-\frac{\sqrt{3}}{3}}-\frac{\sqrt{3}}{2}\]
Simplify \(\sqrt{3}\sqrt{3}\) to \(\sqrt{9}\).
\[\frac{1+\frac{\sqrt{9}}{3}}{\sqrt{3}-\frac{\sqrt{3}}{3}}-\frac{\sqrt{3}}{2}\]
Since \(3\times 3=9\), the square root of \(9\) is \(3\).
\[\frac{1+\frac{3}{3}}{\sqrt{3}-\frac{\sqrt{3}}{3}}-\frac{\sqrt{3}}{2}\]
Cancel \(3\).
\[\frac{1+1}{\sqrt{3}-\frac{\sqrt{3}}{3}}-\frac{\sqrt{3}}{2}\]
Simplify \(1+1\) to \(2\).
\[\frac{2}{\sqrt{3}-\frac{\sqrt{3}}{3}}-\frac{\sqrt{3}}{2}\]
Simplify \(\sqrt{3}-\frac{\sqrt{3}}{3}\) to \(\frac{2\sqrt{3}}{3}\).
\[\frac{2}{\frac{2\sqrt{3}}{3}}-\frac{\sqrt{3}}{2}\]
Invert and multiply.
\[2\times \frac{3}{2\sqrt{3}}-\frac{\sqrt{3}}{2}\]
Cancel \(2\).
\[\frac{3}{\sqrt{3}}-\frac{\sqrt{3}}{2}\]
Rationalize the denominator: \(\frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{3\sqrt{3}}{3}\).
\[\frac{3\sqrt{3}}{3}-\frac{\sqrt{3}}{2}\]
Cancel \(3\).
\[\sqrt{3}-\frac{\sqrt{3}}{2}\]
Simplify.
\[\frac{\sqrt{3}}{2}\]
Decimal Form: 0.866025
sqrt(3)/2
