Simplify. Rationalize the denominator. 7/-9 - sqrt 5 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator. The conjugate of a complex number a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of -9 - sqrt(5) is -9 + 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. (7 * (-9 + sqrt(5))) / ((-9 - sqrt(5)) * (-9 + 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 our denominator: (-9 - sqrt(5)) * (-9 + sqrt(5)) = (-9)^2 - (sqrt(5))^2 = 81 - 5 = 76Distribute Numerator: Distribute the numerator. Now we distribute 7 across the conjugate in the numerator: 7 * (-9) + 7 * sqrt(5) = -63 + 7sqrt(5)Write Simplified Expression: Write the simplified expression. The expression is now simplified to: (-63 + 7sqrt(5)) / 76

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Simplify. Rationalize the denominator. $\newline$ $\frac{7}{-9 - \sqrt{5}}$

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Algebra 1

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{7}{-9 - \sqrt{5}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of a complex number $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-9 - \sqrt{5}$ is $-9 + \sqrt{5}$ .
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. $\newline$ $(7 \cdot (-9 + \sqrt{5})) / ((-9 - \sqrt{5}) \cdot (-9 + \sqrt{5}))$
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 our denominator: $\newline$ $(-9 - \sqrt{5}) * (-9 + \sqrt{5}) = (-9)^2 - (\sqrt{5})^2 = 81 - 5 = 76$
Distribute Numerator: Distribute the numerator. $\newline$ Now we distribute $7$ across the conjugate in the numerator: $\newline$ $7 \times (-9) + 7 \times \sqrt{5} = -63 + 7\sqrt{5}$
Write Simplified Expression: Write the simplified expression. $\newline$ The expression is now simplified to: $\newline$ $(-63 + 7\sqrt{5}) / 76$