Simplify. Rationalize the denominator. 9/-5 + sqrt 2 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator -5 + sqrt 2. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of -5 + sqrt 2 is -5 - 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. The expression becomes (9 * (-5 - sqrt 2)) / ((-5 + sqrt 2) * (-5 - sqrt 2)).Simplify Numerator Distribution: Simplify the numerator by distributing the multiplication. Simplify 9 * (-5 - sqrt 2): = 9 * (-5) + 9 * (-sqrt 2) = -45 - 9sqrt(2)Simplify Denominator: Simplify the denominator using the difference of squares formula. Simplify (-5 + sqrt 2) * (-5 - sqrt 2): = (-5)^2 - (sqrt 2)^2 = 25 - 2 = 23Write Simplified Expression: Write the simplified expression. The simplified expression is (-45 - 9sqrt(2)) / 23. This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{-5 + \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{-5 + \sqrt{2}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator $-5 + \sqrt{2}$ . The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-5 + \sqrt{2}$ is $-5 - \sqrt{2}$ .
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. $\newline$ The expression becomes $(9 \times (-5 - \sqrt{2})) / ((-5 + \sqrt{2}) \times (-5 - \sqrt{2}))$ .
Simplify Numerator Distribution: Simplify the numerator by distributing the multiplication. $\newline$ Simplify $9 \times (-5 - \sqrt{2})$ : $\newline$ $= 9 \times (-5) + 9 \times (-\sqrt{2})$ $\newline$ $= -45 - 9\sqrt{2}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ Simplify $(-5 + \sqrt{2}) * (-5 - \sqrt{2})$ : $\newline$ = $(-5)^2 - (\sqrt{2})^2$ $\newline$ = $25 - 2$ $\newline$ = $23$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression is $(-45 - 9\sqrt{2}) / 23$ . $\newline$ This fraction is already in simplest form.