Simplify. Rationalize the denominator. 3/6 + sqrt 3 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator. 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. To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. (3 * (6 - sqrt(3))) / ((6 + sqrt(3)) * (6 - sqrt(3)))Simplify Numerator: Simplify the numerator. Multiply 3 by each term in the conjugate. 3 * 6 - 3 * sqrt(3) = 18 - 3 * sqrt(3)Simplify Denominator: Simplify the denominator. Use the difference of squares formula: (a + b)(a - b) = a^2 - b^2. (6)^2 - (sqrt(3))^2 = 36 - 3 = 33Write Simplified Expression: Write the simplified expression. Place the simplified numerator over the simplified denominator. (18 - 3 * sqrt(3)) / 33Simplify Fraction: Simplify the fraction by dividing each term in the numerator by the denominator. 18 / 33 - (3 * sqrt(3)) / 33 = 6/11 - sqrt(3)/11

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

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Algebra 2

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. $\newline$ $\frac{18}{33} - \frac{3 \cdot \sqrt{3}}{33}$ $\newline$ = $\frac{6}{11} - \frac{\sqrt{3}}{11}$