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

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Find Conjugate: Select the conjugate of $-5 - \sqrt{5}$ . $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . $\newline$ So, the conjugate of $-5 - \sqrt{5}$ is $-5 + \sqrt{5}$ .
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 $(-5 + \sqrt{5})/(-5 + \sqrt{5})$ . $\newline$ This gives us $(4 \cdot (-5 + \sqrt{5}))/((-5 - \sqrt{5}) \cdot (-5 + \sqrt{5}))$ .
Simplify Numerator: Simplify the numerator by distributing the $4$ . $4 \times (-5 + \sqrt{5})$ gives us $-20 + 4 \times \sqrt{5}$ .
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ $(-5 - \sqrt{5}) * (-5 + \sqrt{5})$ is equal to $(-5)^2 - (\sqrt{5})^2$ . $\newline$ This simplifies to $25 - 5$ , which equals $20$ .
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression is $(-20 + 4 \times \sqrt{5})/20$ .
Final Simplification: Simplify the expression by dividing each term in the numerator by the denominator. $\newline$ $-\frac{20}{20}$ simplifies to $-1$ , and $\frac{4 \cdot \sqrt{5}}{20}$ simplifies to $\frac{\sqrt{5}}{5}$ . $\newline$ So the final simplified expression is $-1 + \frac{\sqrt{5}}{5}$ .

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