Simplify. Rationalize the denominator. 10/2 + sqrt 5 ______

Select Conjugate: Select the conjugate of 2 + sqrt(5). Conjugate of a a + sqrt(b): a - sqrt(b) Conjugate of 2 + sqrt(5): 2 - sqrt(5)Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator. 10/(2 + sqrt(5)) * (2 - sqrt(5))/(2 - sqrt(5))Simplify Numerator: Simplify the numerator by distributing the 10 to both terms in the conjugate. 10 * (2 - sqrt(5)) = 10 * 2 - 10 * sqrt(5) = 20 - 10 * sqrt(5)Simplify Denominator: Simplify the denominator by using the difference of squares formula. (2 + sqrt(5)) * (2 - sqrt(5)) = 2^2 - (sqrt(5))^2 = 4 - 5 = -1Combine Numerator and Denominator: Combine the simplified numerator and denominator. (20 - 10 * sqrt(5))/(-1)Final Simplified Expression: Since dividing by -1 simply changes the sign of the numerator, we can write the final simplified expression. -20 + 10 * sqrt(5)

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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{2 + \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{2 + \sqrt{5}}

Select Conjugate: Select the conjugate of $2 + \sqrt{5}$ . Conjugate of $a + \sqrt{b}$ : $a - \sqrt{b}$ Conjugate of $2 + \sqrt{5}$ : $2 - \sqrt{5}$
Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator. $\newline$ $\frac{10}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}}$
Simplify Numerator: Simplify the numerator by distributing the $10$ to both terms in the conjugate. $10 \times (2 - \sqrt{5})$ $= 10 \times 2 - 10 \times \sqrt{5}$ $= 20 - 10 \times \sqrt{5}$
Simplify Denominator: Simplify the denominator by using the difference of squares formula. $\newline$ $(2 + \sqrt{5}) \times (2 - \sqrt{5})$ $\newline$ $= 2^2 - (\sqrt{5})^2$ $\newline$ $= 4 - 5$ $\newline$ $= -1$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ $(20 - 10 \cdot \sqrt{5})/(-1)$
Final Simplified Expression: Since dividing by $-1$ simply changes the sign of the numerator, we can write the final simplified expression. $-20 + 10 \times \sqrt{5}$