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Simplify. Rationalize the denominator. $\newline$ $\frac{2}{-6 - \sqrt{2}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{2}{-6 - \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 $-6 - \sqrt{2}$ is $-6 + \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and 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 in both the numerator and the denominator. $\newline$ $(\frac{2}{-6 - \sqrt{2}}) \times (\frac{-6 + \sqrt{2}}{-6 + \sqrt{2}})$
Apply Distributive Property: Apply the distributive property to the numerator. $\newline$ Multiply $2$ by each term in the conjugate $(-6 + \sqrt{2})$ . $\newline$ $2 \times (-6) + 2 \times \sqrt{2} = -12 + 2\sqrt{2}$
Apply Difference of Squares: Apply the difference of squares to the denominator. $\newline$ $(-6 - \sqrt{2}) * (-6 + \sqrt{2}) = (-6)^2 - (\sqrt{2})^2$ $\newline$ $36 - 2 = 34$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. The expression now is $(-12 + 2\sqrt{2}) / 34$ .
Simplify by Dividing: Simplify the expression by dividing each term in the numerator by the denominator. $\frac{-12}{34}$ + $\frac{2\sqrt{2}}{34}$
Reduce Fractions: Reduce the fractions to their simplest form. $\newline$ Both $-12$ and $34$ are divisible by $2$ , so we can simplify the first term to $-\frac{6}{17}$ . The second term cannot be simplified further. $\newline$ $\left(-\frac{6}{17}\right) + \left(\frac{\sqrt{2}}{17}\right)$

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