Simplify. Rationalize the denominator. 9/-5 + sqrt 3 ______

Identify Conjugate: Select the conjugate of -5 + sqrt(3). 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. To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. (9 * (-5 - sqrt(3)))/((-5 + sqrt(3)) * (-5 - sqrt(3)))Simplify Numerator: Simplify the numerator. Now we distribute the 9 in the numerator across the conjugate. 9 * (-5) + 9 * (-sqrt(3)) = -45 - 9sqrt(3)Simplify Denominator: Simplify the denominator. We use the difference of squares formula, which states that (a + b)(a - b) = a^2 - b^2. (-5)^2 - (sqrt(3))^2 = 25 - 3 = 22Write Simplified Expression: Write the simplified expression. Now we have the simplified numerator and denominator. (-45 - 9sqrt(3))/22 This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{-5 + \sqrt{3}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{-5 + \sqrt{3}}

Identify Conjugate: Select the conjugate of $-5 + \sqrt{3}$ . $\newline$ 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 numerator and the denominator by the conjugate of the denominator. $\newline$ $(9 \times (-5 - \sqrt{3}))/((-5 + \sqrt{3}) \times (-5 - \sqrt{3}))$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $9$ in the numerator across the conjugate. $\newline$ $9 \times (-5) + 9 \times (-\sqrt{3}) = -45 - 9\sqrt{3}$
Simplify Denominator: Simplify the denominator. $\newline$ We use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ . $\newline$ $(-5)^2 - (\sqrt{3})^2 = 25 - 3 = 22$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and denominator. $\newline$ $(-45 - 9\sqrt{3})/22$ $\newline$ This fraction is already in simplest form.