Simplify. Rationalize the denominator. 9/9 - sqrt 2 ______

Identify Conjugate: Identify the conjugate of the denominator. The conjugate of a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of 9 - sqrt(2) is 9 + sqrt(2).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the expression by (9 + sqrt(2))/(9 + sqrt(2)). (9/(9 - sqrt(2))) * ((9 + sqrt(2))/(9 + sqrt(2)))Apply Distributive Property: Apply the distributive property to the numerator. Multiply 9 by each term in the conjugate (9 + sqrt(2)). 9 * 9 + 9 * sqrt(2) = 81 + 9sqrt(2)Apply Difference of Squares: Apply the difference of squares formula to the denominator. (9 - sqrt(2)) * (9 + sqrt(2)) = 9^2 - (sqrt(2))^2 = 81 - 2Simplify Denominator: Simplify the denominator. 81 - 2 = 79Write Simplified Expression: Write the simplified expression. The rationalized expression is (81 + 9sqrt(2))/79.

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

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Algebra 1

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{9}{9 - \sqrt{2}}

Identify Conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $9 - \sqrt{2}$ is $9 + \sqrt{2}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the expression by $(9 + \sqrt{2})/(9 + \sqrt{2})$ . $\newline$ $(9/(9 - \sqrt{2})) \times ((9 + \sqrt{2})/(9 + \sqrt{2}))$
Apply Distributive Property: Apply the distributive property to the numerator. $\newline$ Multiply $9$ by each term in the conjugate $(9 + \sqrt{2})$ . $\newline$ $9 \times 9 + 9 \times \sqrt{2} = 81 + 9\sqrt{2}$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ $(9 - \sqrt{2}) \times (9 + \sqrt{2}) = 9^2 - (\sqrt{2})^2 = 81 - 2$
Simplify Denominator: Simplify the denominator. $81 - 2 = 79$
Write Simplified Expression: Write the simplified expression. $\newline$ The rationalized expression is $(81 + 9\sqrt{2})/79$ .