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Simplify. Rationalize the denominator. $\newline$ $\frac{8}{-5 + \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{8}{-5 + \sqrt{3}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator $-5 + \sqrt{3}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-5 + \sqrt{3}$ is $-5 - \sqrt{3}$ .
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. $\newline$ The expression becomes: $(8 \cdot (-5 - \sqrt{3}))/((-5 + \sqrt{3}) \cdot (-5 - \sqrt{3}))$ .
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $8$ by each term in the conjugate $-5 - \sqrt{3}$ : $\newline$ $8 \times (-5) + 8 \times (-\sqrt{3})$ $\newline$ = $-40 - 8\sqrt{3}$ .
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ $(-5 + \sqrt{3}) * (-5 - \sqrt{3})$ is a difference of squares which simplifies to: $\newline$ $(-5)^2 - (\sqrt{3})^2$ $\newline$ $= 25 - 3$ $\newline$ $= 22$ .
Write Simplified Expression: Write the simplified expression. $\newline$ The fraction now is $(-40 - 8\sqrt{3})/22$ .
Simplify Fraction: Simplify the fraction by dividing each term in the numerator by the denominator. $\newline$ $-\frac{40}{22}$ simplifies to $-\frac{20}{11}$ and $-\frac{8\sqrt{3}}{22}$ simplifies to $-\frac{4\sqrt{3}}{11}$ . $\newline$ So the final expression is $-\frac{20}{11} - \frac{4\sqrt{3}}{11}$ .

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