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(c) $\newline$ $\frac{1+\sqrt{5}}{1-\sqrt{5}}$

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Multiply radical expressions

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Q. (c)

\newline

\frac{1+\sqrt{5}}{1-\sqrt{5}}

Multiply by Conjugate: First, let's multiply the numerator and the denominator by the conjugate of the denominator to get rid of the square root in the denominator. $\newline$ So we multiply $(1+\sqrt{5})/(1-\sqrt{5})$ by $(1+\sqrt{5})/(1+\sqrt{5})$ .
Perform Multiplication: Now, let's do the multiplication. $\newline$ Numerator: $(1+\sqrt{5}) \times (1+\sqrt{5}) = 1^2 + 2\times1\times\sqrt{5} + (\sqrt{5})^2$ . $\newline$ Denominator: $(1-\sqrt{5}) \times (1+\sqrt{5}) = 1^2 - (\sqrt{5})^2$ .
Simplify Expressions: Simplify the expressions we just found. $\newline$ Numerator: $1 + 2\sqrt{5} + 5$ . $\newline$ Denominator: $1 - 5$ .
Combine Like Terms: Combine like terms in the numerator and denominator. $\newline$ Numerator: $6 + 2\sqrt{5}$ . $\newline$ Denominator: $-4$ .
Divide Numerator by Denominator: Now, divide each term in the numerator by the denominator. $\newline$ $(\frac{6}{-4}) + (\frac{2\sqrt{5}}{-4})$ .
Simplify Fractions: Simplify the fractions. $\newline$ $-\frac{3}{2} - \left(\frac{\sqrt{5}}{2}\right)$ .

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