Question

simplify. rationalize the denominator.
\\(\frac{-2}{3 + \sqrt{3}}\\)

Explanation:

Step1: Identify conjugate of denominator

The denominator is \(3 + \sqrt{3}\), its conjugate is \(3 - \sqrt{3}\). Multiply numerator and denominator by \(3 - \sqrt{3}\).
$$\frac{-2}{3 + \sqrt{3}} \times \frac{3 - \sqrt{3}}{3 - \sqrt{3}}$$

Step2: Multiply numerators and denominators

Numerator: \(-2\times(3 - \sqrt{3}) = -6 + 2\sqrt{3}\)
Denominator: \((3 + \sqrt{3})(3 - \sqrt{3}) = 3^2 - (\sqrt{3})^2 = 9 - 3 = 6\)
So we have \(\frac{-6 + 2\sqrt{3}}{6}\)

Step3: Simplify the fraction

Factor out 2 from numerator: \(\frac{2(-3 + \sqrt{3})}{6} = \frac{-3 + \sqrt{3}}{3}\) or rewrite as \(\frac{\sqrt{3}-3}{3}\)

Answer:

\(\frac{\sqrt{3} - 3}{3}\)