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

Find Conjugate: Select the conjugate of 2 - sqrt(5). Conjugate of a number in the form a - sqrt(b) is a + sqrt(b). So, the conjugate of 2 - sqrt(5) is 2 + sqrt(5).Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator. The expression is 10/(2 - sqrt(5)). We multiply by (2 + sqrt(5))/(2 + sqrt(5)) to rationalize the denominator. 10/(2 - sqrt(5)) * (2 + sqrt(5))/(2 + sqrt(5))Simplify Numerator: Simplify the numerator by distributing the 10. 10 * (2 + sqrt(5)) = 10 * 2 + 10 * sqrt(5) = 20 + 10 * sqrt(5)Simplify Denominator: Simplify the denominator 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) Since dividing by -1 just changes the sign, we get: -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}}

Find Conjugate: Select the conjugate of $2 - \sqrt{5}$ . Conjugate of a number in the form $a - \sqrt{b}$ is $a + \sqrt{b}$ . So, the conjugate of $2 - \sqrt{5}$ is $2 + \sqrt{5}$ .
Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator. $\newline$ The expression is $\frac{10}{2 - \sqrt{5}}$ . $\newline$ We multiply by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ 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 $10$ . $10 \times (2 + \sqrt{5})$ $= 10 \times 2 + 10 \times \sqrt{5}$ $= 20 + 10 \times \sqrt{5}$
Simplify Denominator: Simplify the denominator 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. $\newline$ $(20 + 10 \cdot \sqrt{5})/(-1)$ $\newline$ Since dividing by $-1$ just changes the sign, we get: $\newline$ $-20 - 10 \cdot \sqrt{5}$