Simplify. Rationalize the denominator. 4/-8 - sqrt 5 ______

Select Conjugate: Select the conjugate of -8 - sqrt(5). Conjugate of a - sqrt(b): a + sqrt(b) Conjugate of -8 - sqrt(5): -8 + sqrt(5)Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. 4/(-8 - sqrt(5)) * (-8 + sqrt(5))/(-8 + sqrt(5))Simplify Numerator: Simplify the numerator by distributing the 4 across the conjugate -8 + sqrt(5). 4 * (-8) + 4 * sqrt(5) = -32 + 4sqrt(5)Simplify Denominator: Simplify the denominator by using the difference of squares formula: (a - b)(a + b) = a^2 - b^2. (-8)^2 - (sqrt(5))^2 = 64 - 5 = 59Combine Numerator and Denominator: Combine the simplified numerator and denominator. (-32 + 4sqrt(5))/59 This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{-8 - \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{-8 - \sqrt{5}}

Select Conjugate: Select the conjugate of $-8 - \sqrt{5}$ . $\newline$ Conjugate of $a - \sqrt{b}$ : $a + \sqrt{b}$ $\newline$ Conjugate of $-8 - \sqrt{5}$ : $-8 + \sqrt{5}$
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ $\frac{4}{(-8 - \sqrt{5})} \times \frac{(-8 + \sqrt{5})}{(-8 + \sqrt{5})}$
Simplify Numerator: Simplify the numerator by distributing the $4$ across the conjugate $-8 + \sqrt{5}$ . $4 \times (-8) + 4 \times \sqrt{5}$ $= -32 + 4\sqrt{5}$
Simplify Denominator: Simplify the denominator by using the difference of squares formula: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(-8)^2 - (\sqrt{5})^2$ $\newline$ = $64$ - $5$ $\newline$ = $59$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(-32 + 4\sqrt{5})/59$ $\newline$ This fraction is already in simplest form.