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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{-7 - \sqrt{5}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{-7 - \sqrt{5}}

Identify Conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-7 - \sqrt{5}$ is $-7 + \sqrt{5}$ .
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. $\newline$ $\frac{10 \times (-7 + \sqrt{5})}{(-7 - \sqrt{5}) \times (-7 + \sqrt{5})}$
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: $\newline$ $(-7)^2 - (\sqrt{5})^2 = 49 - 5 = 44$
Distribute Numerator: Distribute the numerator. $\newline$ Now we distribute $10$ across the conjugate in the numerator: $\newline$ $10 \times (-7) + 10 \times \sqrt{5} = -70 + 10\sqrt{5}$
Combine Results: Combine the results to write the final answer. $\newline$ The rationalized expression is: $\newline$ $(-70 + 10\sqrt{5}) / 44$
Simplify Expression: Simplify the expression if possible. $\newline$ We can simplify the expression by dividing both terms in the numerator by the denominator: $\newline$ $(-\frac{70}{44}) + (\frac{10\sqrt{5}}{44})$ $\newline$ This simplifies to: $\newline$ $(-\frac{35}{22}) + (\frac{5\sqrt{5}}{22})$

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