Simplify. Rationalize the denominator. (–4)/(9 – sqrt 2) ______

Multiply by conjugate: To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $9 - \sqrt{2}$ is $9 + \sqrt{2}$.Simplify numerator: Multiply the numerator and the denominator by the conjugate of the denominator: $(-4) \times (9 + \sqrt{2})$ over $(9 - \sqrt{2}) \times (9 + \sqrt{2})$.Simplify denominator: Simplify the numerator by distributing $-4$ to both terms in the conjugate: $-4 \times 9$ and $-4 \times \sqrt{2}$, which gives us $-36 - 4\sqrt{2}$.Calculate denominator: Simplify the denominator by using the difference of squares formula: $(a - b)(a + b) = a^2 - b^2$. Here, $a = 9$ and $b = \sqrt{2}$, so we get $9^2 - (\sqrt{2})^2$, which simplifies to $81 - 2$.Combine numerator and denominator: Calculate the simplified denominator: $81 - 2 = 79$.Combine the simplified numerator and denominator to get the final answer: $(-36 - 4\sqrt{2}) / 79$.

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

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\frac{-4}{9 - \sqrt{2}}

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______

Multiply by conjugate: To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $9 - \sqrt{2}$ is $9 + \sqrt{2}$ .
Simplify numerator: Multiply the numerator and the denominator by the conjugate of the denominator: $(-4) \times (9 + \sqrt{2})$ over $(9 - \sqrt{2}) \times (9 + \sqrt{2})$ .
Simplify denominator: Simplify the numerator by distributing $-4$ to both terms in the conjugate: $-4 \times 9$ and $-4 \times \sqrt{2}$ , which gives us $-36 - 4\sqrt{2}$ .
Calculate denominator: Simplify the denominator by using the difference of squares formula: $(a - b)(a + b) = a^2 - b^2$ . Here, $a = 9$ and $b = \sqrt{2}$ , so we get $9^2 - (\sqrt{2})^2$ , which simplifies to $81 - 2$ .
Combine numerator and denominator: Calculate the simplified denominator: $81 - 2 = 79$ .
Combine numerator and denominator: Calculate the simplified denominator: $81 - 2 = 79$ .Combine the simplified numerator and denominator to get the final answer: $(-36 - 4\sqrt{2}) / 79$ .