Simplify. Rationalize the denominator. 6/-6 - sqrt 3 ______

Identify Conjugate: Identify the conjugate of the denominator. The conjugate of a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of -6 - sqrt(3) is -6 + sqrt(3).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the expression by a fraction equivalent to 1 that has the conjugate of the denominator as both its numerator and denominator. (6 / (-6 - sqrt(3))) * ((-6 + sqrt(3)) / (-6 + sqrt(3)))Apply Distributive Property: Apply the distributive property to multiply the numerators and the denominators. Numerator: 6 * (-6 + sqrt(3)) = -36 + 6*sqrt(3) Denominator: (-6 - sqrt(3)) * (-6 + sqrt(3)) = (-6)^2 - (sqrt(3))^2 = 36 - 3Simplify Numerator and Denominator: Simplify the expressions obtained in the numerator and the denominator. Numerator: -36 + 6*sqrt(3) Denominator: 36 - 3 = 33 So the expression becomes (-36 + 6*sqrt(3)) / 33Simplify Fraction: Simplify the fraction by dividing both the terms in the numerator by the denominator. -36 / 33 + (6*sqrt(3)) / 33 = -12/11 + (2*sqrt(3)) / 11

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Simplify. Rationalize the denominator. $\newline$ $\frac{6}{-6 - \sqrt{3}}$

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Algebra 1

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{6}{-6 - \sqrt{3}}

Identify Conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-6 - \sqrt{3}$ is $-6 + \sqrt{3}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the expression by a fraction equivalent to $1$ that has the conjugate of the denominator as both its numerator and denominator. $\newline$ $\frac{6}{-6 - \sqrt{3}}$ * $\frac{-6 + \sqrt{3}}{-6 + \sqrt{3}}$
Apply Distributive Property: Apply the distributive property to multiply the numerators and the denominators. $\newline$ Numerator: $6 \times (-6 + \sqrt{3}) = -36 + 6\times\sqrt{3}$ $\newline$ Denominator: $(-6 - \sqrt{3}) \times (-6 + \sqrt{3}) = (-6)^2 - (\sqrt{3})^2 = 36 - 3$
Simplify Numerator and Denominator: Simplify the expressions obtained in the numerator and the denominator. $\newline$ Numerator: $-36 + 6\sqrt{3}$ $\newline$ Denominator: $36 - 3 = 33$ $\newline$ So the expression becomes $\frac{-36 + 6\sqrt{3}}{33}$
Simplify Fraction: Simplify the fraction by dividing both the terms in the numerator by the denominator. $\newline$ $-\frac{36}{33} + \frac{6\sqrt{3}}{33}$ $\newline$ = $-\frac{12}{11} + \frac{2\sqrt{3}}{11}$