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

Identify Conjugate of Denominator: Identify the conjugate of the denominator -3 + sqrt(2). The conjugate of a + sqrt(b) is a - sqrt(b), so the conjugate of -3 + sqrt(2) is -3 - 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 (-3 - sqrt(2))/(-3 - sqrt(2)). 2/(-3 + sqrt(2)) * (-3 - sqrt(2))/(-3 - sqrt(2))Distribute Numerator: Distribute the numerator. Multiply 2 by each term in the conjugate -3 - sqrt(2). 2 * (-3) - 2 * sqrt(2) = -6 - 2sqrt(2)Expand Denominator: Expand the denominator using the difference of squares formula. (-3 + sqrt(2)) * (-3 - sqrt(2)) is a difference of squares which simplifies to (-3)^2 - (sqrt(2))^2. 9 - 2 = 7Write Simplified Expression: Write the simplified expression. The simplified expression with a rationalized denominator is (-6 - 2sqrt(2))/7.

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate of Denominator: Identify the conjugate of the denominator $-3 + \sqrt{2}$ . $\newline$ The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$ , so the conjugate of $-3 + \sqrt{2}$ is $-3 - \sqrt{2}$ .
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 (-3 - \sqrt{2})/(-3 - \sqrt{2})$\newline$. $\newline$ $\frac{2}{-3 + \sqrt{2}} \times \frac{-3 - \sqrt{2}}{-3 - \sqrt{2}}$
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $2$ by each term in the conjugate $-3 - \sqrt{2}$ . $\newline$ $2 \times (-3) - 2 \times \sqrt{2}$ $\newline$ $= -6 - 2\sqrt{2}$
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ $(-3 + \sqrt{2}) * (-3 - \sqrt{2})$ is a difference of squares which simplifies to $(-3)^2 - (\sqrt{2})^2$ . $\newline$ $9 - 2$ $\newline$ $= 7$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression with a rationalized denominator is $(-6 - 2\sqrt{2})/7$ .