Simplify. Rationalize the denominator. 10/-9 - sqrt 2 ______

Identify Conjugate: Identify the conjugate of the denominator. The conjugate of 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. 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. (10/(-9 - sqrt(2))) * ((-9 + sqrt(2))/(-9 + sqrt(2)))Apply Distributive Property: Apply the distributive property to multiply the numerators and the denominators. Numerator: 10 * (-9 + sqrt(2)) = -90 + 10*sqrt(2) Denominator: (-9 - sqrt(2)) * (-9 + sqrt(2)) = (-9)^2 - (sqrt(2))^2 = 81 - 2Simplify Denominator: Simplify the denominator. 81 - 2 = 79Write Simplified Expression: Write the simplified expression. The simplified expression with a rationalized denominator is (-90 + 10*sqrt(2)) / 79.

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

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Algebra 1

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ 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. $\newline$ $\frac{10}{-9 - \sqrt{2}}$ * $\frac{-9 + \sqrt{2}}{-9 + \sqrt{2}}$
Apply Distributive Property: Apply the distributive property to multiply the numerators and the denominators. $\newline$ Numerator: $10 \times (-9 + \sqrt{2}) = -90 + 10\times\sqrt{2}$ $\newline$ Denominator: $(-9 - \sqrt{2}) \times (-9 + \sqrt{2}) = (-9)^2 - (\sqrt{2})^2 = 81 - 2$
Simplify Denominator: Simplify the denominator. $81 - 2 = 79$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression with a rationalized denominator is $(-90 + 10\sqrt{2}) / 79$ .