Simplify. Rationalize the denominator. 3/-6 - sqrt 2 ______

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

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

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Algebra 1

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: $\newline$ $(-18 + 3\sqrt{2}) / 34$