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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{-9 - \sqrt{2}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{-9 - \sqrt{2}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-9 - \sqrt{2}$ is $-9 + \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the fraction by a form of $1$ that will eliminate the square root in the denominator. This form of $1$ is the conjugate of the denominator over itself. $\newline$ $(\frac{9}{-9 - \sqrt{2}}) \times (\frac{-9 + \sqrt{2}}{-9 + \sqrt{2}})$
Multiply Numerator: Perform the multiplication in the numerator. $\newline$ Multiply $9$ by the conjugate of the denominator. $\newline$ $9 \times (-9 + \sqrt{2}) = -81 + 9\sqrt{2}$
Multiply Denominator: Perform the multiplication in the denominator. $\newline$ Multiply the denominator by its conjugate. This is a difference of squares. $\newline$ $(-9 - \sqrt{2}) * (-9 + \sqrt{2}) = (-9)^2 - (\sqrt{2})^2 = 81 - 2$
Simplify Denominator: Simplify the denominator. $\newline$ Calculate the difference in the denominator. $\newline$ $81 - 2 = 79$
Write Simplified Expression: Write the simplified expression. $\newline$ Combine the simplified numerator and denominator. $\newline$ $(-81 + 9\sqrt{2}) / 79$

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