Simplify. Rationalize the denominator. $\newline$ $\frac{6}{-9 - \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{6}{-9 - \sqrt{5}}

Identify conjugate of denominator: Identify the conjugate of the denominator $-9 - \sqrt{5}$ . $\newline$ The conjugate of a number of the form $a - \sqrt{b}$ 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 the numerator and the denominator by the conjugate of the denominator. $\newline$ So, we have $(6 \times (-9 + \sqrt{5}))/((-9 - \sqrt{5}) \times (-9 + \sqrt{5}))$ .
Simplify numerator: Simplify the numerator. $\newline$ Now, we distribute the $6$ across the conjugate in the numerator: $\newline$ $6 \times (-9) + 6 \times \sqrt{5}$ $\newline$ $= -54 + 6 \times \sqrt{5}$ .
Simplify denominator: Simplify the denominator using the difference of squares formula. $\newline$ The denominator simplifies as follows: $\newline$ $(-9 - \sqrt{5}) * (-9 + \sqrt{5})$ $\newline$ $= (-9)^2 - (\sqrt{5})^2$ $\newline$ $= 81 - 5$ $\newline$ $= 76$ .
Write simplified expression: Write the simplified expression. $\newline$ Now we have the simplified expression: $\newline$ $(-54 + 6 \times \sqrt{5})/76$ .
Simplify fraction: Simplify the fraction by dividing each term in the numerator by the denominator. $\newline$ $-\frac{54}{76} + \frac{6 \cdot \sqrt{5}}{76}$ $\newline$ We can simplify $-\frac{54}{76}$ by dividing both numerator and denominator by $2$ : $\newline$ $-\frac{27}{38} + \frac{6 \cdot \sqrt{5}}{76}$ .
Check second term: Check if the second term can be simplified. $\newline$ The second term $(6 \times \sqrt{5})/76$ can be simplified by dividing both numerator and denominator by $2$ : $\newline$ $3 \times \sqrt{5}/38$ .
Combine simplified terms: Combine the simplified terms. $\newline$ The final simplified expression is: $\newline$ $-\frac{27}{38} + \frac{3 \cdot \sqrt{5}}{38}$ .