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

Identify Conjugate of Denominator: Identify the conjugate of the denominator 2 - sqrt(2). The conjugate of a number of the form a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of 2 - sqrt(2) is 2 + 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. (2 * (2 + sqrt(2))) / ((2 - sqrt(2)) * (2 + sqrt(2)))Simplify Numerator: Simplify the numerator. Multiply 2 by each term in the conjugate. 2 * 2 + 2 * sqrt(2) = 4 + 2sqrt(2)Simplify Denominator: Simplify the denominator using the difference of squares formula. (2 - sqrt(2)) * (2 + sqrt(2)) = 2^2 - (sqrt(2))^2 = 4 - 2 = 2Write Simplified Expression: Write the simplified expression. The simplified expression is (4 + 2sqrt(2)) / 2.Divide Numerator by Denominator: Simplify the expression by dividing each term in the numerator by the denominator. (4/2) + (2sqrt(2)/2) = 2 + sqrt(2)

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate of Denominator: Identify the conjugate of the denominator $2 - \sqrt{2}$ . $\newline$ The conjugate of a number of the form $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $2 - \sqrt{2}$ is $2 + \sqrt{2}$ .
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. $\newline$ $\frac{2 \times (2 + \sqrt{2})}{(2 - \sqrt{2}) \times (2 + \sqrt{2})}$
Simplify Numerator: Simplify the numerator. $\newline$ Multiply $2$ by each term in the conjugate. $\newline$ $2 \times 2 + 2 \times \sqrt{2}$ $\newline$ = $4 + 2\sqrt{2}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ $(2 - \sqrt{2}) \times (2 + \sqrt{2}) = 2^2 - (\sqrt{2})^2$ $\newline$ $= 4 - 2$ $\newline$ $= 2$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression is $(4 + 2\sqrt{2}) / 2$ .
Divide Numerator by Denominator: Simplify the expression by dividing each term in the numerator by the denominator. $\newline$ $(\frac{4}{2}) + (\frac{2\sqrt{2}}{2})$ $\newline$ = $2 + \sqrt{2}$