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Simplify. Rationalize the denominator. $\newline$ $\frac{3}{-6 - \sqrt{2}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{3}{-6 - \sqrt{2}}

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 $-6 - \sqrt{2}$ is $-6 + \sqrt{2}$ .
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. $\newline$ $(3 \times (-6 + \sqrt{2})) / ((-6 - \sqrt{2}) \times (-6 + \sqrt{2}))$
Apply Distributive Property: Apply the distributive property to the numerator. $\newline$ Multiply $3$ by each term in the conjugate. $\newline$ $3 \times (-6) + 3 \times \sqrt{2} = -18 + 3\sqrt{2}$
Apply Difference of Squares: Apply the difference of squares to the denominator. $\newline$ When we multiply two conjugates, the result is the difference of squares. $\newline$ $(-6 - \sqrt{2}) \times (-6 + \sqrt{2}) = (-6)^2 - (\sqrt{2})^2 = 36 - 2$
Simplify Denominator: Simplify the denominator. $\newline$ Subtract $2$ from $36$ to get the simplified denominator. $\newline$ $36 - 2 = 34$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression with the rationalized denominator is: $\newline$ $(-18 + 3\sqrt{2}) / 34$

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