Simplify. Rationalize the denominator. 3/-2 - sqrt 5 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator. The conjugate of a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of -2 - sqrt(5) is -2 + sqrt(5).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. (3 * (-2 + sqrt(5))) / ((-2 - sqrt(5)) * (-2 + sqrt(5)))Apply Difference of Squares: Apply the difference of squares formula to the denominator. The difference of squares formula is (a - b)(a + b) = a^2 - b^2. Applying this to the denominator we get: (-2)^2 - (sqrt(5))^2 = 4 - 5 = -1.Distribute Numerator: Distribute the numerator. Multiply 3 by each term in the conjugate. 3 * (-2) + 3 * sqrt(5) = -6 + 3sqrt(5).Combine and Simplify: Combine the results and simplify. Since the denominator simplifies to -1, we divide the entire numerator by -1 to get the final simplified form. (-6 + 3sqrt(5)) / -1 = 6 - 3sqrt(5).

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Simplify. Rationalize the denominator. $\newline$ $\frac{3}{-2 - \sqrt{5}}$

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Algebra 1

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{3}{-2 - \sqrt{5}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-2 - \sqrt{5}$ is $-2 + \sqrt{5}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. $\newline$ $\frac{3 \times (-2 + \sqrt{5})}{(-2 - \sqrt{5}) \times (-2 + \sqrt{5})}$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ The difference of squares formula is $(a - b)(a + b) = a^2 - b^2$ . Applying this to the denominator we get: $\newline$ $(-2)^2 - (\sqrt{5})^2 = 4 - 5 = -1$ .
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $3$ by each term in the conjugate. $\newline$ $3 \times (-2) + 3 \times \sqrt{5} = -6 + 3\sqrt{5}.$
Combine and Simplify: Combine the results and simplify. $\newline$ Since the denominator simplifies to $-1$ , we divide the entire numerator by $-1$ to get the final simplified form. $\newline$ $(-6 + 3\sqrt{5}) / -1 = 6 - 3\sqrt{5}$ .