Simplify. Rationalize the denominator. 8/-2 + sqrt 5 ______

Select Conjugate: Select the conjugate of -2 + sqrt 5. Conjugate of a - sqrt(b): a + sqrt(b) Conjugate of -2 + sqrt 5: -2 - sqrt 5Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the expression by (-2 - sqrt 5)/(-2 - sqrt 5). 8/(-2 + sqrt 5) * (-2 - sqrt 5)/(-2 - sqrt 5)Simplify Numerator: Simplify the numerator by distributing the multiplication. 8 * (-2 - sqrt 5) = 8 * (-2) + 8 * (-sqrt 5) = -16 - 8sqrt(5)Simplify Denominator: Simplify the denominator using the difference of squares formula. (-2 + sqrt 5) * (-2 - sqrt 5) = (-2)^2 - (sqrt 5)^2 = 4 - 5 = -1Combine Numerator and Denominator: Combine the simplified numerator and denominator. (-16 - 8sqrt(5)) / (-1) When we divide by -1, we change the sign of the numerator. = 16 + 8sqrt(5)

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Simplify. Rationalize the denominator. $\newline$ $\frac{8}{-2 + \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{8}{-2 + \sqrt{5}}

Select Conjugate: Select the conjugate of $-2 + \sqrt{5}$ . $\newline$ Conjugate of $a - \sqrt{b}$ : $a + \sqrt{b}$ $\newline$ Conjugate of $-2 + \sqrt{5}$ : $-2 - \sqrt{5}$
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the expression by $(-2 - \sqrt{5})/(-2 - \sqrt{5})$ . $\newline$ $\frac{8}{-2 + \sqrt{5}} \times \frac{-2 - \sqrt{5}}{-2 - \sqrt{5}}$
Simplify Numerator: Simplify the numerator by distributing the multiplication. $\newline$ $8 \times (-2 - \sqrt{5})$ $\newline$ = $8 \times (-2) + 8 \times (-\sqrt{5})$ $\newline$ = $-16 - 8\sqrt{5}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ $(-2 + \sqrt{5}) * (-2 - \sqrt{5})$ $\newline$ = $(-2)^2 - (\sqrt{5})^2$ $\newline$ = $4 - 5$ $\newline$ = $-1$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(-16 - 8\sqrt{5}) / (-1)$ $\newline$ When we divide by $-1$ , we change the sign of the numerator. $\newline$ $= 16 + 8\sqrt{5}$