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

Find Conjugate: Select the conjugate of -6 + sqrt(5). Conjugate of a - sqrt(b): a + sqrt(b) Conjugate of -6 + sqrt(5): -6 - sqrt(5)Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. 5/(-6 + sqrt(5)) * (-6 - sqrt(5))/(-6 - sqrt(5))Simplify Numerator: Simplify the numerator. Multiply 5 by the conjugate -6 - sqrt(5). 5 * (-6) - 5 * sqrt(5) = -30 - 5sqrt(5)Simplify Denominator: Simplify the denominator. Simplify (-6 + sqrt(5)) * (-6 - sqrt(5)). (-6)^2 - (sqrt(5))^2 = 36 - 5 = 31Combine Numerator and Denominator: Combine the simplified numerator and denominator. (-30 - 5sqrt(5))/31 This fraction is already in simplest form.

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Find Conjugate: Select the conjugate of $-6 + \sqrt{5}$ . $\newline$ Conjugate of $a - \sqrt{b}$ : $a + \sqrt{b}$ $\newline$ Conjugate of $-6 + \sqrt{5}$ : $-6 - \sqrt{5}$
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. $\newline$ $\frac{5}{-6 + \sqrt{5}} \times \frac{-6 - \sqrt{5}}{-6 - \sqrt{5}}$
Simplify Numerator: Simplify the numerator. $\newline$ Multiply $5$ by the conjugate $-6 - \sqrt{5}$ . $\newline$ $5 \times (-6) - 5 \times \sqrt{5}$ $\newline$ $= -30 - 5\sqrt{5}$
Simplify Denominator: Simplify the denominator. $\newline$ Simplify $(-6 + \sqrt{5}) * (-6 - \sqrt{5})$ . $\newline$ $(-6)^2 - (\sqrt{5})^2$ $\newline$ = $36 - 5$ $\newline$ = $31$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(-30 - 5\sqrt{5})/31$ $\newline$ This fraction is already in simplest form.