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Simplify. Rationalize the denominator. $\newline$ $\frac{3}{6 + \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{3}{6 + \sqrt{3}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $6 + \sqrt{3}$ is $6 - \sqrt{3}$ .
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. $\newline$ $\frac{3 \times (6 - \sqrt{3})}{(6 + \sqrt{3}) \times (6 - \sqrt{3})}$
Simplify Numerator: Simplify the numerator. $\newline$ Multiply $3$ by each term in the conjugate. $\newline$ $3 \times 6 - 3 \times \sqrt{3}$ $\newline$ = $18 - 3 \times \sqrt{3}$
Simplify Denominator: Simplify the denominator. $\newline$ Use the difference of squares formula: $(a + b)(a - b) = a^2 - b^2$ . $\newline$ $(6)^2 - (\sqrt{3})^2$ $\newline$ = $36$ - $3$ $\newline$ = $33$
Write Simplified Expression: Write the simplified expression. $\newline$ Place the simplified numerator over the simplified denominator. $\newline$ $(18 - 3 \cdot \sqrt{3}) / 33$
Simplify Fraction: Simplify the fraction by dividing each term in the numerator by the denominator. $\newline$ $\frac{18}{33} - \frac{3 \cdot \sqrt{3}}{33}$ $\newline$ = $\frac{6}{11} - \frac{\sqrt{3}}{11}$

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