Simplify. Rationalize the denominator. 4/2 + sqrt 5 ______

Identify Conjugate: Identify the conjugate of the denominator. The conjugate of a binomial in the form of a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of the denominator 2 + sqrt(5) is 2 - sqrt(5).Multiply by Conjugate Fraction: Multiply the original expression by a fraction equivalent to 1 that has the conjugate of the denominator as both the numerator and the denominator. To rationalize the denominator, we multiply the original expression by (2 - sqrt(5))/(2 - sqrt(5)). (4/(2 + sqrt(5))) * ((2 - sqrt(5))/(2 - sqrt(5)))Distribute Numerator: Distribute the numerator. Multiply 4 by each term in the conjugate (2 - sqrt(5)). 4 * 2 - 4 * sqrt(5) = 8 - 4sqrt(5)Apply Difference of Squares: Apply the difference of squares formula to the denominator. (2 + sqrt(5))(2 - sqrt(5)) is a difference of squares, which simplifies to: 2^2 - (sqrt(5))^2 = 4 - 5 = -1Combine Results: Combine the results of Step 3 and Step 4 to get the final expression. (8 - 4sqrt(5)) / (-1) When dividing by -1, we change the sign of the numerator. -8 + 4sqrt(5)

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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{2 + \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{2 + \sqrt{5}}

Identify Conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of a binomial in the form of $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of the denominator $2 + \sqrt{5}$ is $2 - \sqrt{5}$ .
Multiply by Conjugate Fraction: Multiply the original expression by a fraction equivalent to $1$ that has the conjugate of the denominator as both the numerator and the denominator. $\newline$ To rationalize the denominator, we multiply the original expression by $(2 - \sqrt{5})/(2 - \sqrt{5})$ . $\newline$ $(4/(2 + \sqrt{5})) \cdot ((2 - \sqrt{5})/(2 - \sqrt{5}))$
Distribute Numerator: Distribute the numerator. $\newline$ Multiply $4$ by each term in the conjugate $(2 - \sqrt{5})$ . $\newline$ $4 \times 2 - 4 \times \sqrt{5}$ $\newline$ $= 8 - 4\sqrt{5}$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ $(2 + \sqrt{5})(2 - \sqrt{5})$ is a difference of squares, which simplifies to: $\newline$ $2^2 - (\sqrt{5})^2$ $\newline$ = $4 - 5$ $\newline$ = $-1$
Combine Results: Combine the results of Step $3$ and Step $4$ to get the final expression. $\newline$ $(8 - 4\sqrt{5}) / (-1)$ $\newline$ When dividing by $-1$ , we change the sign of the numerator. $\newline$ $-8 + 4\sqrt{5}$