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Simplify. Rationalize the denominator. $\newline$ $\frac{7}{-9 - \sqrt{5}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{7}{-9 - \sqrt{5}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of a complex number $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, 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. $\newline$ $(7 \cdot (-9 + \sqrt{5})) / ((-9 - \sqrt{5}) \cdot (-9 + \sqrt{5}))$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ The difference of squares formula is $(a - b)(a + b) = a^2 - b^2$ . Applying this to our denominator: $\newline$ $(-9 - \sqrt{5}) * (-9 + \sqrt{5}) = (-9)^2 - (\sqrt{5})^2 = 81 - 5 = 76$
Distribute Numerator: Distribute the numerator. $\newline$ Now we distribute $7$ across the conjugate in the numerator: $\newline$ $7 \times (-9) + 7 \times \sqrt{5} = -63 + 7\sqrt{5}$
Write Simplified Expression: Write the simplified expression. $\newline$ The expression is now simplified to: $\newline$ $(-63 + 7\sqrt{5}) / 76$

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