Simplify. Rationalize the denominator. 2/-7 + sqrt 2 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator -7 + sqrt 2. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b), and vice versa. Therefore, the conjugate of -7 + sqrt 2 is -7 - 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 a fraction equivalent to 1 that has the conjugate of the denominator as both its numerator and denominator. (2/(-7 + sqrt 2)) * ((-7 - sqrt 2)/(-7 - sqrt 2))Distribute Numerator: Distribute the numerator. Multiply 2 by each term in the conjugate -7 - sqrt 2. 2 * (-7) + 2 * (-sqrt 2) = -14 - 2sqrt(2)Expand Denominator: Expand the denominator using the difference of squares formula. (-7 + sqrt 2) * (-7 - sqrt 2) = (-7)^2 - (sqrt 2)^2 = 49 - 2 = 47Write Simplified Expression: Write the simplified expression. Now we have the numerator as -14 - 2sqrt(2) and the denominator as 47. So the simplified expression is (-14 - 2sqrt(2))/47.

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Simplify. Rationalize the denominator. $\newline$ $\frac{2}{-7 + \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{2}{-7 + \sqrt{2}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator $-7 + \sqrt{2}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ , and vice versa. Therefore, the conjugate of $-7 + \sqrt{2}$ is $-7 - \sqrt{2}$ .
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 a fraction equivalent to $1$ that has the conjugate of the denominator as both its numerator and denominator. $\newline$ $\frac{2}{-7 + \sqrt{2}}$ * $\frac{-7 - \sqrt{2}}{-7 - \sqrt{2}}$
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $2$ by each term in the conjugate $-7 - \sqrt{2}$ . $\newline$ $2 \times (-7) + 2 \times (-\sqrt{2})$ $\newline$ $= -14 - 2\sqrt{2}$
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ (\(-7 + \sqrt{ $2$ }) * ( $-7$ - \sqrt{ $2$ }) = ( $-7$ )^ $2$ - (\sqrt{ $2$ })^ $2$ $\newline$ = $49$ - $2$ $\newline$ = $47$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the numerator as $-14 - 2\sqrt{2}$ and the denominator as $47$ . $\newline$ So the simplified expression is $\frac{-14 - 2\sqrt{2}}{47}$ .