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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{-9 - \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{-9 - \sqrt{3}}

Identify conjugate of denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-9 - \sqrt{3}$ is $-9 + \sqrt{3}$ .
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 consists of the conjugate of the denominator over itself. $\newline$ $\frac{10}{-9 - \sqrt{3}}$ * $\frac{-9 + \sqrt{3}}{-9 + \sqrt{3}}$
Apply distributive property: Apply the distributive property (foil method) to the numerator. $\newline$ Multiply $10$ by each term in the conjugate $(-9 + \sqrt{3})$ . $\newline$ $10 \times (-9) + 10 \times \sqrt{3} = -90 + 10\sqrt{3}$
Simplify numerator: Apply the distributive property (foil method) to the denominator. $\newline$ Multiply each term in $(-9 - \sqrt{3})$ by each term in $(-9 + \sqrt{3})$ . $\newline$ $(-9 \times -9) + (-9 \times \sqrt{3}) - (\sqrt{3} \times -9) - (\sqrt{3} \times \sqrt{3})$
Apply distributive property: Simplify the denominator. $\newline$ Calculate each multiplication in the denominator. $\newline$ $81 - 9\sqrt{3} + 9\sqrt{3} - 3$ $\newline$ The middle terms cancel each other out, so we are left with: $\newline$ $81 - 3 = 78$
Simplify denominator: Write the simplified expression. $\newline$ The numerator is $-90 + 10\sqrt{3}$ , and the denominator is $78$ . $\newline$ $\frac{-90 + 10\sqrt{3}}{78}$
Write simplified expression: Simplify the fraction by dividing each term in the numerator by the denominator. $\newline$ Divide $-90$ by $78$ and $10\sqrt{3}$ by $78$ . $\newline$ $-90/78 = -45/39$ (simplifying by dividing both numerator and denominator by $2$ ) $\newline$ $10\sqrt{3}/78 = 5\sqrt{3}/39$ (simplifying by dividing both numerator and denominator by $2$ )
Simplify fraction: Combine the simplified terms. $\newline$ The final simplified expression is: $\newline$ $-\frac{45}{39} + \frac{5\sqrt{3}}{39}$

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