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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{4 + \sqrt{5}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{4 + \sqrt{5}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of a number in the form of $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $4 + \sqrt{5}$ is $4 - \sqrt{5}$ .
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. $\newline$ $\frac{4}{4 + \sqrt{5}}$ $\cdot$ $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $4$ by the conjugate of the denominator. $\newline$ $4 \times (4 - \sqrt{5}) = 16 - 4\sqrt{5}$
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ $(4 + \sqrt{5}) \times (4 - \sqrt{5}) = 4^2 - (\sqrt{5})^2$ $\newline$ $= 16 - 5$ $\newline$ $= 11$
Write Simplified Expression: Write the simplified expression. $\newline$ Place the simplified numerator over the simplified denominator. $\newline$ $(16 - 4\sqrt{5})/11$ $\newline$ This is the expression with the denominator rationalized.

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