Simplify. Rationalize the denominator. 4/-7 - 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 -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. (4 * (-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 our denominator: (-7 - sqrt(5)) * (-7 + sqrt(5)) = (-7)^2 - (sqrt(5))^2 = 49 - 5 = 44Distribute Numerator: Distribute the numerator. Now we distribute 4 across the conjugate in the numerator: 4 * (-7) + 4 * sqrt(5) = -28 + 4sqrt(5)Combine Simplified Form: Combine the results to express the simplified form. The expression now reads as: (-28 + 4sqrt(5)) / 44Divide by Denominator: Simplify the expression by dividing both terms in the numerator by the denominator. Both terms in the numerator can be divided by 44 to simplify the expression further: -28/44 + (4sqrt(5))/44Reduce to Simplest Form: Reduce the fractions to their simplest form. -28/44 reduces to -7/11 and 4/44 reduces to 1/11, so the expression simplifies to: -7/11 + (sqrt(5))/11

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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{-7 - \sqrt{5}}$

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Algebra 1

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{-7 - \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 $-7 - \sqrt{5}$ 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. $\newline$ $(4 \cdot (-7 + \sqrt{5})) / ((-7 - \sqrt{5}) \cdot (-7 + \sqrt{5}))$
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 our denominator: $\newline$ $(-7 - \sqrt{5}) * (-7 + \sqrt{5}) = (-7)^2 - (\sqrt{5})^2 = 49 - 5 = 44$
Distribute Numerator: Distribute the numerator. $\newline$ Now we distribute $4$ across the conjugate in the numerator: $\newline$ $4 \times (-7) + 4 \times \sqrt{5} = -28 + 4\sqrt{5}$
Combine Simplified Form: Combine the results to express the simplified form. $\newline$ The expression now reads as: $\newline$ $(-28 + 4\sqrt{5}) / 44$
Divide by Denominator: Simplify the expression by dividing both terms in the numerator by the denominator. Both terms in the numerator can be divided by $44$ to simplify the expression further: $\frac{-28}{44} + \frac{4\sqrt{5}}{44}$
Reduce to Simplest Form: Reduce the fractions to their simplest form. $\newline$ $-\frac{28}{44}$ reduces to $-\frac{7}{11}$ and $\frac{4}{44}$ reduces to $\frac{1}{11}$ , so the expression simplifies to: $\newline$ $-\frac{7}{11} + \frac{\sqrt{5}}{11}$