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

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate: Identify the conjugate of the denominator $5 - \sqrt{5}$ . $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . $\newline$ So, the conjugate of $5 - \sqrt{5}$ is $5 + \sqrt{5}$ .
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 $(5 + \sqrt{5})/(5 + \sqrt{5})$ . $\newline$ This gives us $(10 \cdot (5 + \sqrt{5}))/((5 - \sqrt{5}) \cdot (5 + \sqrt{5}))$ .
Simplify Numerator: Simplify the numerator by distributing the multiplication. $\newline$ $10 \times (5 + \sqrt{5})$ equals $10 \times 5 + 10 \times \sqrt{5}$ , which simplifies to $50 + 10 \times \sqrt{5}$ .
Simplify Denominator: Simplify the denominator by using the difference of squares formula. $\newline$ $(5 - \sqrt{5}) \times (5 + \sqrt{5})$ equals $5^2 - (\sqrt{5})^2$ , which simplifies to $25 - 5$ , resulting in $20$ .
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression is $(\frac{50 + 10 \times \sqrt{5}}{20})$ .
Further Simplify Expression: Simplify the expression further by dividing each term in the numerator by the denominator. $\frac{50}{20} + \frac{(10 \cdot \sqrt{5})}{20}$ simplifies to $2.5 + 0.5 \cdot \sqrt{5}$ .

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