Simplify. Rationalize the denominator. 9/-2 + sqrt 3 ______

Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. (9 * (-2 - sqrt 3)) / ((-2 + sqrt 3) * (-2 - sqrt 3))Simplify Numerator: Simplify the numerator by distributing 9 to both terms in the conjugate. 9 * (-2) + 9 * (-sqrt 3) = -18 - 9sqrt 3Simplify Denominator: Simplify the denominator by using the difference of squares formula, which is (a - b)(a + b) = a^2 - b^2. (-2)^2 - (sqrt 3)^2 = 4 - 3 = 1Combine Numerator and Denominator: Combine the simplified numerator and denominator. (-18 - 9sqrt 3) / 1 Since dividing by 1 does not change the value, the expression simplifies to: -18 - 9sqrt 3

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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{-2 + \sqrt{3}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{-2 + \sqrt{3}}

Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ $(9 \times (-2 - \sqrt{3})) / ((-2 + \sqrt{3}) \times (-2 - \sqrt{3}))$
Simplify Numerator: Simplify the numerator by distributing $9$ to both terms in the conjugate. $9 \times (-2) + 9 \times (-\sqrt{3}) = -18 - 9\sqrt{3}$
Simplify Denominator: Simplify the denominator by using the difference of squares formula, which is $(a - b)(a + b) = a^2 - b^2.$ $\newline$ $(-2)^2 - (\sqrt{3})^2$ $\newline$ $= 4 - 3$ $\newline$ $= 1$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(-18 - 9\sqrt{3}) / 1$ $\newline$ Since dividing by $1$ does not change the value, the expression simplifies to: $\newline$ $-18 - 9\sqrt{3}$