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Simplify. Rationalize the denominator. $\newline$ $\frac{3}{-7 - \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{3}{-7 - \sqrt{3}}

Select Conjugate: Select the conjugate of $-7 - \sqrt{3}$ . $\newline$ The conjugate of a number of the form $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-7 - \sqrt{3}$ is $-7 + \sqrt{3}$ .
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: $\newline$ $(3 \times (-7 + \sqrt{3})) / ((-7 - \sqrt{3}) \times (-7 + \sqrt{3}))$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $3$ in the numerator across the conjugate: $\newline$ $3 \times (-7) + 3 \times \sqrt{3}$ $\newline$ $= -21 + 3\sqrt{3}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ The denominator is in the form of $(a - b)(a + b)$ , which simplifies to $a^2 - b^2$ : $\newline$ $(-7)^2 - (\sqrt{3})^2$ $\newline$ $= 49 - 3$ $\newline$ $= 46$
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator over the simplified denominator: $\newline$ $(-21 + 3\sqrt{3}) / 46$ $\newline$ This fraction is already in simplest form.

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