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

Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. (3 * (-3 - sqrt(3)))/((-3 + sqrt(3)) * (-3 - sqrt(3)))Simplify numerator: Simplify the numerator by distributing the 3. 3 * (-3) + 3 * (-sqrt(3)) = -9 - 3sqrt(3)Simplify denominator: Simplify the denominator by using the difference of squares formula (a - b)(a + b) = a^2 - b^2. (-3)^2 - (sqrt(3))^2 = 9 - 3 = 6Combine numerator and denominator: Combine the simplified numerator and denominator. (-9 - 3sqrt(3))/6Divide by denominator: Divide each term in the numerator by the denominator. -9/6 - (3sqrt(3))/6Simplify terms: Simplify each term. -3/2 - (sqrt(3))/2

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ $(3 \times (-3 - \sqrt{3}))/((-3 + \sqrt{3}) \times (-3 - \sqrt{3}))$
Simplify numerator: Simplify the numerator by distributing the $3$ . $3 \times (-3) + 3 \times (-\sqrt{3}) = -9 - 3\sqrt{3}$
Simplify denominator: Simplify the denominator by using the difference of squares formula $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(-3)^2 - (\sqrt{3})^2 = 9 - 3 = 6$
Combine numerator and denominator: Combine the simplified numerator and denominator. $(-9 - 3\sqrt{3})/6$
Divide by denominator: Divide each term in the numerator by the denominator. $\newline$ $-\frac{9}{6} - \frac{3\sqrt{3}}{6}$
Simplify terms: Simplify each term. $\newline$ $-\frac{3}{2} - \frac{\sqrt{3}}{2}$