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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{-9 - \sqrt{5}}$

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Simplify radical expressions using conjugates

Full solution

Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{-9 - \sqrt{5}}

Select Conjugate: Select the conjugate of $-9 - \sqrt{5}$ . $\newline$ Conjugate of $a - \sqrt{b}$ : $a + \sqrt{b}$ $\newline$ Conjugate of $-9 - \sqrt{5}$ : $-9 + \sqrt{5}$
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. $\newline$ $(9 \times (-9 + \sqrt{5})) / ((-9 - \sqrt{5}) \times (-9 + \sqrt{5}))$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ (\(-9 - \sqrt{ $5$ }) * ( $-9$ + \sqrt{ $5$ }) = ( $-9$ )^ $2$ - (\sqrt{ $5$ })^ $2$ $\newline$ = $81$ - $5$ $\newline$ = $76$
Distribute Numerator: Distribute the numerator. $\newline$ $9 \times (-9 + \sqrt{5}) = 9 \times (-9) + 9 \times \sqrt{5}$ $\newline$ $= -81 + 9 \times \sqrt{5}$
Combine Results: Combine the results of Step $3$ and Step $4$ to get the final answer. $(-81 + 9 \times \sqrt{5}) / 76$

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