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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{-9 + \sqrt{2}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{-9 + \sqrt{2}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator $-9 + \sqrt{2}$ . $\newline$ 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. $\newline$ To rationalize the denominator, we multiply the fraction by a form of $1$ that consists of the conjugate of the denominator over itself. $\newline$ $\frac{10}{-9 + \sqrt{2}}$ * $\frac{-9 - \sqrt{2}}{-9 - \sqrt{2}}$
Distribute Multiplication in Numerator: Distribute the multiplication in the numerator. $\newline$ Multiply $10$ by each term in the conjugate $-9 - \sqrt{2}$ . $\newline$ $10 \times (-9) = -90$ $\newline$ $10 \times (-\sqrt{2}) = -10\sqrt{2}$ $\newline$ So the numerator becomes $-90 - 10\sqrt{2}$ .
Apply Difference of Squares: Apply the difference of squares in the denominator. $\newline$ When we multiply the conjugate pair, we get $(-9 + \sqrt{2}) \times (-9 - \sqrt{2})$ , which is a difference of squares. $\newline$ $(-9)^2 - (\sqrt{2})^2$ $\newline$ = $81$ - $2$ $\newline$ = $79$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the numerator as $-90 - 10\sqrt{2}$ and the denominator as $79$ . $\newline$ So the simplified expression is $\frac{-90 - 10\sqrt{2}}{79}$ .

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