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Simplify. Rationalize the denominator.

(-2)/(8+sqrt5)

Simplify. Rationalize the denominator. $\newline$ $\frac{-2}{8+\sqrt{5}}$

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Simplify radical expressions using conjugates

Full solution

Q. Simplify. Rationalize the denominator.

\newline

\frac{-2}{8+\sqrt{5}}

Select Conjugate: Select the conjugate of the denominator to rationalize it. $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . $\newline$ Therefore, the conjugate of $8 + \sqrt{5}$ is $8 - \sqrt{5}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ This will eliminate the square root from the denominator. $\newline$ $(-2) \times (8 - \sqrt{5})$ / $(8 + \sqrt{5}) \times (8 - \sqrt{5})$
Use Difference of Squares: Use the difference of squares formula to simplify the denominator. $\newline$ $(a + b)(a - b) = a^2 - b^2$ $\newline$ $(8 + \sqrt{5})(8 - \sqrt{5}) = 8^2 - (\sqrt{5})^2$ $\newline$ $= 64 - 5$ $\newline$ $= 59$
Simplify Numerator: Simplify the numerator by distributing $-2$ to both terms in the conjugate. $\newline$ (-2) \times (8 - \sqrt{5}) = (-2) \times 8 + (-2) \times (-\sqrt{5})\(\newline= -16 + 2\sqrt{5}\)
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(-16 + 2\sqrt{5}) / 59$ $\newline$ This is the simplified form of the original expression with a rationalized denominator.

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