Simplify. Rationalize the denominator. 3/-6 + sqrt 5 ______

Identify Conjugate: Identify the conjugate of the denominator -6 + sqrt 5. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of -6 + sqrt 5 is -6 - sqrt 5.Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the fraction by a form of 1 that consists of the conjugate of the denominator over itself. (3/(-6 + sqrt 5)) * ((-6 - sqrt 5)/(-6 - sqrt 5))Distribute Numerator: Distribute the numerator. Multiply 3 by each term in the conjugate -6 - sqrt 5. 3 * (-6) + 3 * (-sqrt 5) = -18 - 3sqrt(5)Expand Denominator: Expand the denominator using the difference of squares formula. (-6 + sqrt 5) * (-6 - sqrt 5) = (-6)^2 - (sqrt 5)^2 = 36 - 5 = 31Write Simplified Expression: Write the simplified expression. The fraction with the rationalized denominator is (-18 - 3sqrt(5))/31. This fraction is already in its simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{3}{-6 + \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{3}{-6 + \sqrt{5}}

Identify Conjugate: Identify the conjugate of the denominator $-6 + \sqrt{5}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-6 + \sqrt{5}$ is $-6 - \sqrt{5}$ .
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. $\newline$ $\frac{3}{-6 + \sqrt{5}}$ * $\frac{-6 - \sqrt{5}}{-6 - \sqrt{5}}$
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $3$ by each term in the conjugate $-6 - \sqrt{5}$ . $\newline$ $3 \times (-6) + 3 \times (-\sqrt{5})$ $\newline$ $= -18 - 3\sqrt{5}$
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ (-6 + \sqrt{5}) * (-6 - \sqrt{5}) = (-6)^2 - (\sqrt{5})^2$\newline= 36 - 5 $\newline$ = 31$
Write Simplified Expression: Write the simplified expression. $\newline$ The fraction with the rationalized denominator is $(-18 - 3\sqrt{5})/31$ . $\newline$ This fraction is already in its simplest form.