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

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate: Select the conjugate of $-9 + \sqrt{5}$ . $\newline$ The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$ , so the conjugate of $-9 + \sqrt{5}$ is $-9 - \sqrt{5}$ .
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: $\newline$ $(5 \cdot (-9 - \sqrt{5})) / ((-9 + \sqrt{5}) \cdot (-9 - \sqrt{5}))$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $5$ across the terms in the conjugate: $\newline$ $5 \times (-9) + 5 \times (-\sqrt{5})$ $\newline$ $= -45 - 5\sqrt{5}$
Simplify Denominator: Simplify the denominator. $\newline$ We use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ : $\newline$ $(-9)^2 - (\sqrt{5})^2$ $\newline$ $= 81 - 5$ $\newline$ $= 76$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and denominator: $\newline$ $(-45 - 5\sqrt{5}) / 76$ $\newline$ This fraction is already in simplest form.

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