Simplify. Rationalize the denominator. 2/-9 + sqrt 2 ______

Find Conjugate: Select the conjugate of -9 + sqrt 2. 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. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is -9 - sqrt 2. So, we have: (2 * (-9 - sqrt 2)) / ((-9 + sqrt 2) * (-9 - sqrt 2))Simplify Numerator: Simplify the numerator. Now we distribute the 2 in the numerator across the conjugate: 2 * (-9) + 2 * (-sqrt 2) = -18 - 2sqrt(2)Simplify Denominator: Simplify the denominator. We use the difference of squares formula, which states that (a + b)(a - b) = a^2 - b^2. So, we have: (-9)^2 - (sqrt 2)^2 = 81 - 2 = 79Write Final Expression: Write the simplified expression. Now we have the simplified numerator and denominator: (-18 - 2sqrt(2)) / 79 This is the expression with the denominator rationalized.

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{2}{-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.