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Simplify. Rationalize the denominator. $\newline$ $\frac{7}{-4 - \sqrt{5}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{7}{-4 - \sqrt{5}}

Identify conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-4 - \sqrt{5}$ is $-4 + \sqrt{5}$ .
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 a fraction equivalent to $1$ that has the conjugate of the denominator as both its numerator and denominator. $\newline$ $\frac{7}{-4 - \sqrt{5}}$ * $\frac{-4 + \sqrt{5}}{-4 + \sqrt{5}}$
Apply distributive property: Apply the distributive property to multiply the numerators and the denominators. $\newline$ Numerator: $7 \times (-4 + \sqrt{5}) = -28 + 7\sqrt{5}$ $\newline$ Denominator: $(-4 - \sqrt{5}) \times (-4 + \sqrt{5}) = (-4)^2 - (\sqrt{5})^2 = 16 - 5 = 11$
Write simplified expression: Write the simplified expression. $\newline$ The simplified expression with a rationalized denominator is: $\newline$ $(-28 + 7\sqrt{5}) / 11$

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