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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{-9 - \sqrt{2}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{-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 $a + \sqrt{b}$ . 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 expression by a fraction equivalent to $1$ that has the conjugate of the denominator in both the numerator and the denominator. $\newline$ So, we multiply $\frac{4}{-9 - \sqrt{2}}$ by $\frac{-9 + \sqrt{2}}{-9 + \sqrt{2}}$ .
Multiply numerator: Perform the multiplication in the numerator. $\newline$ Multiply $4$ by the conjugate $-9 + \sqrt{2}$ . $\newline$ $4 \times (-9 + \sqrt{2}) = -36 + 4\sqrt{2}$
Multiply denominator: Perform the multiplication in the denominator. $\newline$ Multiply $(-9 - \sqrt{2})$ by $(-9 + \sqrt{2})$ . $\newline$ This is a difference of squares, which is $(a - b)(a + b) = a^2 - b^2$ . $\newline$ So, $(-9)^2 - (\sqrt{2})^2 = 81 - 2 = 79$
Combine multiplication results: Combine the results of the multiplication. $\newline$ Now we have $(-36 + 4\sqrt{2})/79$ . $\newline$ This is the simplified form of the expression with a rationalized denominator.

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