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Simplify. Rationalize the denominator. $\newline$ $-\frac{6}{6 + \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Find Conjugate: Select the conjugate of $6 + \sqrt{3}$ .
Conjugate of $a + \sqrt{b}$ : $a - \sqrt{b}$
Conjugate of $6 + \sqrt{3}$ : $6 - \sqrt{3}$
Rationalize Denominator: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ Expression to rationalize the denominator: $(-6)/(6 + \sqrt{3})$ $\newline$ Multiply by $(6 - \sqrt{3})/(6 - \sqrt{3})$ $\newline$ $(-6 \cdot (6 - \sqrt{3}))/((6 + \sqrt{3}) \cdot (6 - \sqrt{3}))$
Simplify Numerator: Simplify the numerator: $-6 \times (6 - \sqrt{3})$ $\newline$ $-6 \times 6 - (-6) \times \sqrt{3}$ $\newline$ $= -36 + 6\sqrt{3}$
Simplify Denominator: Simplify the denominator: $(6 + \sqrt{3}) * (6 - \sqrt{3})$ Using the difference of squares formula: $a^2 - b^2$ $(6)^2 - (\sqrt{3})^2$ = $36 - 3$ = $33$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(-36 + 6\sqrt{3})/33$ $\newline$ This fraction can be further simplified by dividing both terms in the numerator by $33$ . $\newline$ $(-36/33) + (6\sqrt{3}/33)$ $\newline$ $= -12/11 + (2\sqrt{3}/11)$

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