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

Identify Conjugate: Select the conjugate of -3 + sqrt(5). The conjugate of a number of the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of -3 + sqrt(5) is -3 - 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 * (-3 - sqrt(5)))/((-3 + sqrt(5)) * (-3 - sqrt(5)))Simplify Numerator: Simplify the numerator. Now we distribute the 3 in the numerator across the conjugate. 3 * (-3) + 3 * (-sqrt(5)) = -9 - 3sqrt(5)Simplify Denominator: Simplify the denominator. We use the difference of squares formula, which states that (a + b)(a - b) = a^2 - b^2. (-3)^2 - (sqrt(5))^2 = 9 - 5 = 4Write Simplified Expression: Write the simplified expression. Now we have the simplified numerator and denominator. (-9 - 3sqrt(5))/4 This fraction is already in simplest form.

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate: Select the conjugate of $-3 + \sqrt{5}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-3 + \sqrt{5}$ is $-3 - \sqrt{5}$ .
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. $\newline$ $(3 \cdot (-3 - \sqrt{5}))/((-3 + \sqrt{5}) \cdot (-3 - \sqrt{5}))$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $3$ in the numerator across the conjugate. $\newline$ $3 \times (-3) + 3 \times (-\sqrt{5})$ $\newline$ $= -9 - 3\sqrt{5}$
Simplify Denominator: Simplify the denominator. $\newline$ We use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ . $\newline$ $(-3)^2 - (\sqrt{5})^2$ $\newline$ = $9 - 5$ $\newline$ = $4$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and denominator. $\newline$ $(-9 - 3\sqrt{5})/4$ $\newline$ This fraction is already in simplest form.