Simplify. Rationalize the denominator. 6/2 + sqrt 3 ______

Identify conjugate of denominator: Identify the conjugate of the denominator 2 + sqrt(3). The conjugate of a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of 2 + sqrt(3) is 2 - sqrt(3).Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the fraction by 1 in the form of the conjugate over itself: (2 - sqrt(3))/(2 - sqrt(3)). This gives us (6 * (2 - sqrt(3)))/((2 + sqrt(3)) * (2 - sqrt(3))).Simplify numerator by distributing: Simplify the numerator by distributing the 6. 6 * (2 - sqrt(3)) equals 12 - 6 * sqrt(3).Simplify denominator using formula: Simplify the denominator by using the difference of squares formula. (2 + sqrt(3)) * (2 - sqrt(3)) equals 2^2 - (sqrt(3))^2. This simplifies to 4 - 3, which equals 1.Write simplified fraction: Write the simplified fraction. Since the denominator is now 1, the fraction simplifies to just the numerator: 12 - 6 * sqrt(3).

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Simplify. Rationalize the denominator. $\newline$ $\frac{6}{2 + \sqrt{3}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{6}{2 + \sqrt{3}}

Identify conjugate of denominator: Identify the conjugate of the denominator $2 + \sqrt{3}$ . The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $2 + \sqrt{3}$ is $2 - \sqrt{3}$ .
Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the fraction by $1$ in the form of the conjugate over itself: $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ . $\newline$ This gives us $\frac{6 \times (2 - \sqrt{3})}{(2 + \sqrt{3}) \times (2 - \sqrt{3})}$ .
Simplify numerator by distributing: Simplify the numerator by distributing the $6$ . $\newline$ $6 \times (2 - \sqrt{3})$ equals $12 - 6 \times \sqrt{3}$ .
Simplify denominator using formula: Simplify the denominator by using the difference of squares formula. $\newline$ $(2 + \sqrt{3}) \times (2 - \sqrt{3})$ equals $2^2 - (\sqrt{3})^2$ . $\newline$ This simplifies to $4 - 3$ , which equals $1$ .
Write simplified fraction: Write the simplified fraction. $\newline$ Since the denominator is now $1$ , the fraction simplifies to just the numerator: $12 - 6 \times \sqrt{3}$ .