Simplify. Rationalize the denominator. 9/5 + sqrt 5 ______

Multiply Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. (9 * (5 - sqrt(5))) / ((5 + sqrt(5)) * (5 - sqrt(5)))Simplify Numerator: Simplify the numerator by distributing 9 to both terms in the conjugate. 9 * 5 - 9 * sqrt(5) = 45 - 9 * sqrt(5)Simplify Denominator: Simplify the denominator by using the difference of squares formula, which states that (a + b)(a - b) = a^2 - b^2. (5)^2 - (sqrt(5))^2 = 25 - 5 = 20Combine Terms: Combine the simplified numerator and denominator. (45 - 9 * sqrt(5)) / 20Split Fraction: The expression is now rationalized and simplified. We can split the fraction into two parts to make it clearer. 45/20 - (9 * sqrt(5))/20Reduce Fractions: Simplify each term by reducing the fractions. 45/20 = 9/4 (9 * sqrt(5))/20 = (9/20) * sqrt(5)Final Expression: Write the final simplified and rationalized expression. 9/4 - (9/20) * sqrt(5)

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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{5 + \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{5 + \sqrt{5}}

Multiply Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ $(9 \times (5 - \sqrt{5})) / ((5 + \sqrt{5}) \times (5 - \sqrt{5}))$
Simplify Numerator: Simplify the numerator by distributing $9$ to both terms in the conjugate. $9 \times 5 - 9 \times \sqrt{5}$ $= 45 - 9 \times \sqrt{5}$
Simplify Denominator: Simplify the denominator by using the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ . $(5)^2 - (\sqrt{5})^2$ $= 25 - 5$ $= 20$
Combine Terms: Combine the simplified numerator and denominator. $(45 - 9 \times \sqrt{5}) / 20$
Split Fraction: The expression is now rationalized and simplified. We can split the fraction into two parts to make it clearer. $\frac{45}{20} - \frac{9 \cdot \sqrt{5}}{20}$
Reduce Fractions: Simplify each term by reducing the fractions. $\newline$ $\frac{45}{20} = \frac{9}{4}$ $\newline$ $\frac{9 \cdot \sqrt{5}}{20} = \frac{9}{20} \cdot \sqrt{5}$
Final Expression: Write the final simplified and rationalized expression. $\frac{9}{4} - \left(\frac{9}{20}\right) \cdot \sqrt{5}$