Simplify. Rationalize the denominator. 10/-7 - sqrt 5 ______

Identify Conjugate: Identify the conjugate of the denominator. 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. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. (10 * (-7 + sqrt(5))) / ((-7 - sqrt(5)) * (-7 + 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: (-7)^2 - (sqrt(5))^2 = 49 - 5 = 44Distribute Numerator: Distribute the numerator. Now we distribute 10 across the conjugate in the numerator: 10 * (-7) + 10 * sqrt(5) = -70 + 10sqrt(5)Combine Results: Combine the results to write the final answer. The rationalized expression is: (-70 + 10sqrt(5)) / 44Simplify Expression: Simplify the expression if possible. We can simplify the expression by dividing both terms in the numerator by the denominator: (-70/44) + (10sqrt(5)/44) This simplifies to: (-35/22) + (5sqrt(5)/22)

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

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Algebra 1

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: $\newline$ $(-\frac{35}{22}) + (\frac{5\sqrt{5}}{22})$