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

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Find Conjugate: Select the conjugate of $-9 + \sqrt{2}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ , and vice versa. 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 to rationalize it. $\newline$ To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $-9 - \sqrt{2}$ . $\newline$ So, we have: $\newline$ $\frac{2 \cdot (-9 - \sqrt{2})}{(-9 + \sqrt{2}) \cdot (-9 - \sqrt{2})}$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $2$ in the numerator across the conjugate: $\newline$ $2 \times (-9) + 2 \times (-\sqrt{2})$ $\newline$ $= -18 - 2\sqrt{2}$
Simplify Denominator: Simplify the denominator. $\newline$ We use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ . $\newline$ So, we have: $\newline$ $(-9)^2 - (\sqrt{2})^2$ $\newline$ = $81$ - $2$ $\newline$ = $79$
Write Final Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and denominator: $\newline$ $(-18 - 2\sqrt{2}) / 79$ $\newline$ This is the expression with the denominator rationalized.

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