Simplify. Rationalize the denominator. 9/-8 - sqrt 2 ______

Find Conjugate: Select the conjugate of $-8 - \sqrt{2}$. Conjugate of a number in the form $a - \sqrt{b}$ is $a + \sqrt{b}$. Therefore, the conjugate of $-8 - \sqrt{2}$ is $-8 + \sqrt{2}$.Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the expression by $\frac{-8 + \sqrt{2}}{-8 + \sqrt{2}}$. $\frac{9}{-8 - \sqrt{2}} \times \frac{-8 + \sqrt{2}}{-8 + \sqrt{2}}$Simplify Numerator: Simplify the numerator. Multiply 9 by the conjugate $-8 + \sqrt{2}$. $9 \times (-8) + 9 \times \sqrt{2}$ $-72 + 9\sqrt{2}$Simplify Denominator: Simplify the denominator using the difference of squares formula. $(-8 - \sqrt{2}) \times (-8 + \sqrt{2}) = (-8)^2 - (\sqrt{2})^2$ $64 - 2$ $62$Write Simplified Expression: Write the simplified expression. $\frac{-72 + 9\sqrt{2}}{62}$ This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{-8 - \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{-8 - \sqrt{2}}

Find Conjugate: Select the conjugate of $-8 - \sqrt{2}$ . $\newline$ Conjugate of a number in the form $a - \sqrt{b}$ is $a + \sqrt{b}$ . $\newline$ Therefore, the conjugate of $-8 - \sqrt{2}$ is $-8 + \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the expression by $\frac{-8 + \sqrt{2}}{-8 + \sqrt{2}}$ . $\newline$ $\frac{9}{-8 - \sqrt{2}} \times \frac{-8 + \sqrt{2}}{-8 + \sqrt{2}}$
Simplify Numerator: Simplify the numerator. $\newline$ Multiply $9$ by the conjugate $-8 + \sqrt{2}$ . $\newline$ $9 \times (-8) + 9 \times \sqrt{2}$ $\newline$ $-72 + 9\sqrt{2}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ $(-8 - \sqrt{2}) \times (-8 + \sqrt{2}) = (-8)^2 - (\sqrt{2})^2$ $\newline$ $64 - 2$ $\newline$ $62$
Write Simplified Expression: Write the simplified expression. $\newline$ $\frac{-72 + 9\sqrt{2}}{62}$ $\newline$ This fraction is already in simplest form.