Simplify. Rationalize the denominator. 9/3 + sqrt 2 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of 3 + sqrt(2) is 3 - sqrt(2).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. (9 * (3 - sqrt(2))) / ((3 + sqrt(2)) * (3 - sqrt(2)))Simplify Numerator: Simplify the numerator. Multiply 9 by each term in the conjugate. 9 * 3 - 9 * sqrt(2) = 27 - 9 * sqrt(2)Simplify Denominator: Simplify the denominator. Use the difference of squares formula, which states that (a + b)(a - b) = a^2 - b^2. (3 + sqrt(2)) * (3 - sqrt(2)) = 3^2 - (sqrt(2))^2 = 9 - 2 = 7Write Final Expression: Write the simplified expression. Now that we have simplified both the numerator and the denominator, we can write the final expression. (27 - 9 * sqrt(2)) / 7 This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{9}{3 + \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{3 + \sqrt{2}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $3 + \sqrt{2}$ is $3 - \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ $(9 \times (3 - \sqrt{2})) / ((3 + \sqrt{2}) \times (3 - \sqrt{2}))$
Simplify Numerator: Simplify the numerator. $\newline$ Multiply $9$ by each term in the conjugate. $\newline$ $9 \times 3 - 9 \times \sqrt{2}$ $\newline$ = $27 - 9 \times \sqrt{2}$
Simplify Denominator: Simplify the denominator. $\newline$ Use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ . $\newline$ $(3 + \sqrt{2}) * (3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2$ $\newline$ $= 9 - 2$ $\newline$ $= 7$
Write Final Expression: Write the simplified expression. $\newline$ Now that we have simplified both the numerator and the denominator, we can write the final expression. $\newline$ $(27 - 9 \times \sqrt{2}) / 7$ $\newline$ This fraction is already in simplest form.