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Simplify. Rationalize the denominator. $\newline$ $\frac{5}{-4 - \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{5}{-4 - \sqrt{3}}

Identify conjugate of denominator: Identify the conjugate of the denominator $-4 - \sqrt{3}$ . $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . $\newline$ So, the conjugate of $-4 - \sqrt{3}$ is $-4 + \sqrt{3}$ .
Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the fraction by a form of $1$ that will eliminate the square root in the denominator. This form of $1$ is the conjugate of the denominator over itself. $\newline$ So, we multiply $\frac{5}{-4 - \sqrt{3}}$ by $\frac{-4 + \sqrt{3}}{-4 + \sqrt{3}}$ .
Multiply numerator: Perform the multiplication in the numerator. $\newline$ Multiply $5$ by $(-4 + \sqrt{3})$ . $\newline$ $5 \times (-4 + \sqrt{3}) = -20 + 5\sqrt{3}$
Multiply denominator: Perform the multiplication in the denominator. $\newline$ Multiply $(-4 - \sqrt{3})$ by $(-4 + \sqrt{3})$ . $\newline$ This is a difference of squares, which is $(a - b)(a + b) = a^2 - b^2$ . $\newline$ So, $(-4)^2 - (\sqrt{3})^2 = 16 - 3 = 13$
Write simplified expression: Write the simplified expression. $\newline$ Now we have the numerator as $-20 + 5\sqrt{3}$ and the denominator as $13$ . $\newline$ So, the simplified expression is $\frac{-20 + 5\sqrt{3}}{13}$ .

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