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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{-3 + \sqrt{5}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{-3 + \sqrt{5}}

Identify conjugate of denominator: Identify the conjugate of the denominator $-3 + \sqrt{5}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-3 + \sqrt{5}$ is $-3 - \sqrt{5}$ .
Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the expression by $(-3 - \sqrt{5})/(-3 - \sqrt{5})$ . $\newline$ This gives us $(10 \cdot (-3 - \sqrt{5}))/((-3 + \sqrt{5}) \cdot (-3 - \sqrt{5}))$ .
Simplify numerator by distributing: Simplify the numerator by distributing the multiplication. $\newline$ Multiplying $10$ by each term in the conjugate, we get: $\newline$ $10 \times (-3) - 10 \times \sqrt{5}$ $\newline$ $= -30 - 10\sqrt{5}$
Simplify denominator using formula: Simplify the denominator by using the difference of squares formula. $\newline$ The difference of squares formula states that $(a + b)(a - b) = a^2 - b^2$ . Applying this to our denominator: $\newline$ $(-3)^2 - (\sqrt{5})^2$ $\newline$ $= 9 - 5$ $\newline$ $= 4$
Write simplified expression: Write the simplified expression. $\newline$ Now we have $(-30 - 10\sqrt{5}) / 4$ . To simplify this, we can divide each term in the numerator by the denominator: $\newline$ $(-30/4) - (10\sqrt{5}/4)$ $\newline$ = $-7.5 - 2.5\sqrt{5}$

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