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

Find Conjugate: Select the conjugate of -5 - sqrt(5). Conjugate of a - sqrt(b): a + sqrt(b) Conjugate of -5 - sqrt(5): -5 + sqrt(5)Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. Expression: 6/(-5 - sqrt(5)) Multiply by (-5 + sqrt(5))/(-5 + sqrt(5)) 6 * (-5 + sqrt(5)) / ((-5 - sqrt(5)) * (-5 + sqrt(5)))Simplify Numerator: Simplify the numerator by distributing the 6. 6 * (-5) + 6 * sqrt(5) = -30 + 6 * sqrt(5)Simplify Denominator: Simplify the denominator using the difference of squares formula (a - b)(a + b) = a^2 - b^2. (-5)^2 - (sqrt(5))^2 = 25 - 5 = 20Write Simplified Expression: Write the simplified expression with the rationalized denominator. (-30 + 6 * sqrt(5)) / 20Final Simplification: Simplify the expression by dividing each term in the numerator by the denominator. -30/20 + (6 * sqrt(5))/20 = -3/2 + (3 * sqrt(5))/10

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Find Conjugate: Select the conjugate of $-5 - \sqrt{5}$ . $\newline$ Conjugate of $a - \sqrt{b}$ : $a + \sqrt{b}$ $\newline$ Conjugate of $-5 - \sqrt{5}$ : $-5 + \sqrt{5}$
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ Expression: $\frac{6}{-5 - \sqrt{5}}$ $\newline$ Multiply by $\frac{-5 + \sqrt{5}}{-5 + \sqrt{5}}$ $\newline$ $\frac{6 \times (-5 + \sqrt{5})}{((-5 - \sqrt{5})) \times (-5 + \sqrt{5})}$
Simplify Numerator: Simplify the numerator by distributing the $6$ . $6 \times (-5) + 6 \times \sqrt{5}$ $= -30 + 6 \times \sqrt{5}$
Simplify Denominator: Simplify the denominator using the difference of squares formula $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(-5)^2 - (\sqrt{5})^2$ $\newline$ = $25 - 5$ $\newline$ = $20$
Write Simplified Expression: Write the simplified expression with the rationalized denominator. $(-30 + 6 \cdot \sqrt{5}) / 20$
Final Simplification: Simplify the expression by dividing each term in the numerator by the denominator. $\newline$ $-\frac{30}{20} + \frac{6 \cdot \sqrt{5}}{20}$ $\newline$ = $-\frac{3}{2} + \frac{3 \cdot \sqrt{5}}{10}$