Simplify. Rationalize the denominator. 8/2 - sqrt 5 ______

Select conjugate: Select the conjugate of the denominator 2 - sqrt(5) to rationalize it. The conjugate of a - sqrt(b) is a + sqrt(b). So, the conjugate of 2 - sqrt(5) is 2 + sqrt(5).Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. So, multiply 8/(2 - sqrt(5)) by (2 + sqrt(5))/(2 + sqrt(5)).Multiply numerator: Perform the multiplication in the numerator: 8 * (2 + sqrt(5)). This gives us 16 + 8 * sqrt(5).Multiply denominator: Perform the multiplication in the denominator: (2 - sqrt(5)) * (2 + sqrt(5)). This is a difference of squares, which gives us 2^2 - (sqrt(5))^2.Simplify denominator: Simplify the denominator: 2^2 - (sqrt(5))^2. This gives us 4 - 5.Calculate denominator: Calculate the simplified denominator: 4 - 5. This gives us -1.Combine numerator and denominator: Combine the simplified numerator and denominator: (16 + 8 * sqrt(5)) / -1.Simplify expression: Simplify the expression by dividing both terms in the numerator by the denominator. This gives us -16 - 8 * sqrt(5).

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Select conjugate: Select the conjugate of the denominator $2 - \sqrt{5}$ to rationalize it. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . $\newline$ So, the conjugate of $2 - \sqrt{5}$ is $2 + \sqrt{5}$ .
Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ So, multiply $\frac{8}{2 - \sqrt{5}}$ by $\frac{2 + \sqrt{5}}{2 + \sqrt{5}}$ .
Multiply numerator: Perform the multiplication in the numerator: $8 \times (2 + \sqrt{5})$ . This gives us $16 + 8 \times \sqrt{5}$ .
Multiply denominator: Perform the multiplication in the denominator: $(2 - \sqrt{5}) \times (2 + \sqrt{5})$ . This is a difference of squares, which gives us $2^2 - (\sqrt{5})^2$ .
Simplify denominator: Simplify the denominator: $2^2 - (\sqrt{5})^2$ . This gives us $4 - 5$ .
Calculate denominator: Calculate the simplified denominator: $4 - 5$ . This gives us $-1$ .
Combine numerator and denominator: Combine the simplified numerator and denominator: $(16 + 8 \cdot \sqrt{5}) / -1$ .
Simplify expression: Simplify the expression by dividing both terms in the numerator by the denominator. $\newline$ This gives us $-16 - 8 \times \sqrt{5}$ .