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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{9 + \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{9 + \sqrt{3}}

Select Conjugate: Select the conjugate of $9 + \sqrt{3}$ . $\newline$ Conjugate of $a + \sqrt{b}$ : $a - \sqrt{b}$ $\newline$ Conjugate of $9 + \sqrt{3}$ : $9 - \sqrt{3}$
Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator. $\newline$ $\frac{4}{9 + \sqrt{3}} \times \frac{9 - \sqrt{3}}{9 - \sqrt{3}}$
Simplify Numerator: Simplify the numerator by distributing the $4$ across the conjugate. $4 \times (9 - \sqrt{3})$ $= 4 \times 9 - 4 \times \sqrt{3}$ $= 36 - 4\sqrt{3}$
Simplify Denominator: Simplify the denominator by using the difference of squares formula. $\newline$ $(9 + \sqrt{3}) \times (9 - \sqrt{3})$ $\newline$ $= 9^2 - (\sqrt{3})^2$ $\newline$ $= 81 - 3$ $\newline$ $= 78$
Write with Rationalized Denominator: Write the simplified expression with the rationalized denominator. $\newline$ $(36 - 4\sqrt{3})/78$
Simplify Expression: Simplify the expression by dividing both terms in the numerator by the denominator. $\newline$ $\frac{36}{78} - \frac{4\sqrt{3}}{78}$ $\newline$ = $\frac{18}{39} - \frac{2\sqrt{3}}{39}$ $\newline$ = $\frac{6}{13} - \frac{2\sqrt{3}}{39}$
Check for Further Simplification: Check if the expression can be simplified further. Both terms are in their simplest form, so the expression cannot be simplified further.

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