Simplify. Rationalize the denominator. 5/-4 + sqrt 2 ______

Identify Conjugate: Identify the conjugate of the denominator -4 + sqrt 2. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b), and vice versa. Therefore, the conjugate of -4 + sqrt 2 is -4 - sqrt 2.Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the fraction by a form of 1 that consists of the conjugate of the denominator over itself. (5/(-4 + sqrt 2)) * ((-4 - sqrt 2)/(-4 - sqrt 2))Distribute Numerator: Distribute the numerator. Multiply 5 by each term in the conjugate -4 - sqrt 2. 5 * (-4) + 5 * (-sqrt 2) = -20 - 5sqrt(2)Expand Denominator: Expand the denominator using the difference of squares formula. (-4 + sqrt 2) * (-4 - sqrt 2) = (-4)^2 - (sqrt 2)^2 = 16 - 2 = 14Write Simplified Expression: Write the simplified expression. Place the simplified numerator over the simplified denominator. (-20 - 5sqrt(2)) / 14 This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{5}{-4 + \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{5}{-4 + \sqrt{2}}

Identify Conjugate: Identify the conjugate of the denominator $-4 + \sqrt{2}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ , and vice versa. Therefore, the conjugate of $-4 + \sqrt{2}$ is $-4 - \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the fraction by a form of $1$ that consists of the conjugate of the denominator over itself. $\newline$ $(\frac{5}{-4 + \sqrt{2}}) \times (\frac{-4 - \sqrt{2}}{-4 - \sqrt{2}})$
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $5$ by each term in the conjugate $-4 - \sqrt{2}$ . $\newline$ $5 \times (-4) + 5 \times (-\sqrt{2})$ $\newline$ = $-20 - 5\sqrt{2}$
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ (-4 + \sqrt{2}) * (-4 - \sqrt{2}) = (-4)^2 - (\sqrt{2})^2$\newline= 16 - 2 $\newline$ = 14$
Write Simplified Expression: Write the simplified expression. $\newline$ Place the simplified numerator over the simplified denominator. $\newline$ $(-20 - 5\sqrt{2}) / 14$ $\newline$ This fraction is already in simplest form.