Simplify. Rationalize the denominator. 8/8 + 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 8 + sqrt(3) is 8 - sqrt(3).Multiply by Conjugate Fraction: Multiply the original expression by a fraction equivalent to 1 that has the conjugate of the denominator as both its numerator and denominator. (8/(8 + sqrt(3))) * ((8 - sqrt(3))/(8 - sqrt(3))) = (8 * (8 - sqrt(3)))/((8 + sqrt(3)) * (8 - sqrt(3)))Multiply Numerators: Multiply the numerators together. 8 * (8 - sqrt(3)) = 64 - 8sqrt(3)Multiply Denominators: Multiply the denominators together using the difference of squares formula, a^2 - b^2 = (a + b)(a - b). (8 + sqrt(3)) * (8 - sqrt(3)) = 8^2 - (sqrt(3))^2 = 64 - 3Simplify Denominator: Simplify the denominator. 64 - 3 = 61Write Simplified Expression: Write the simplified expression. (64 - 8sqrt(3))/61

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Simplify. Rationalize the denominator. $\newline$ $\frac{8}{8 + \sqrt{3}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{8}{8 + \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 $8 + \sqrt{3}$ is $8 - \sqrt{3}$ .
Multiply by Conjugate Fraction: Multiply the original expression by a fraction equivalent to $1$ that has the conjugate of the denominator as both its numerator and denominator. $\newline$ $(\frac{8}{8 + \sqrt{3}}) \times \frac{8 - \sqrt{3}}{8 - \sqrt{3}} = \frac{8 \times (8 - \sqrt{3})}{(8 + \sqrt{3}) \times (8 - \sqrt{3})}$
Multiply Numerators: Multiply the numerators together. $\newline$ $8 \times (8 - \sqrt{3}) = 64 - 8\sqrt{3}$
Multiply Denominators: Multiply the denominators together using the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$ . $\newline$ $(8 + \sqrt{3}) \times (8 - \sqrt{3}) = 8^2 - (\sqrt{3})^2 = 64 - 3$
Simplify Denominator: Simplify the denominator. $64 - 3 = 61$
Write Simplified Expression: Write the simplified expression. $(64 - 8\sqrt{3})/61$