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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{-2 - \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Select Conjugate: Select the conjugate of $-2 - \sqrt{3}$ . $\newline$ Conjugate of $a - \sqrt{b}$ : $a + \sqrt{b}$ $\newline$ Conjugate of $-2 - \sqrt{3}$ : $-2 + \sqrt{3}$
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ Expression: $(\frac{4}{-2 - \sqrt{3}}) \cdot (\frac{-2 + \sqrt{3}}{-2 + \sqrt{3}})$
Multiply Numerator: Perform the multiplication in the numerator. $\newline$ Numerator: $4 \times (-2 + \sqrt{3})$ $\newline$ = $4 \times -2 + 4 \times \sqrt{3}$ $\newline$ = $-8 + 4\sqrt{3}$
Multiply Denominator: Perform the multiplication in the denominator using the difference of squares formula. $\newline$ Denominator: $(-2 - \sqrt{3}) \times (-2 + \sqrt{3})$ $\newline$ $= (-2)^2 - (\sqrt{3})^2$ $\newline$ $= 4 - 3$ $\newline$ $= \(1\)
Write Simplified Expression: Write the simplified expression.\(\newline\)Since the denominator is now \(1\), the expression simplifies to the numerator.\(\newline\)Final expression: \(-8 + 4\sqrt{3}\)

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