Simplify. Rationalize the denominator. 5/5 - sqrt 2 ______

Identify Conjugate: Identify the conjugate of the denominator 5 - sqrt(2). The conjugate of a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of 5 - sqrt(2) is 5 + sqrt(2).Multiply by Conjugate: Multiply the original expression by a fraction equivalent to 1 that has the conjugate of the denominator as both its numerator and denominator. To rationalize the denominator, we multiply the original expression by (5 + sqrt(2))/(5 + sqrt(2)). \[ \frac{5}{5 - \sqrt{2}} \times \frac{5 + \sqrt{2}}{5 + \sqrt{2}} \]Multiply in Numerator: Perform the multiplication in the numerator. \[ 5 \times (5 + \sqrt{2}) = 5 \times 5 + 5 \times \sqrt{2} = 25 + 5\sqrt{2} \]Multiply in Denominator: Perform the multiplication in the denominator using the difference of squares formula. \[ (5 - \sqrt{2}) \times (5 + \sqrt{2}) = 5^2 - (\sqrt{2})^2 = 25 - 2 = 23 \]Write Simplified Expression: Write the simplified expression with the rationalized denominator. \[ \frac{25 + 5\sqrt{2}}{23} \] This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{5}{5 - \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{5}{5 - \sqrt{2}}

Identify Conjugate: Identify the conjugate of the denominator $5 - \sqrt{2}$ . $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $5 - \sqrt{2}$ is $5 + \sqrt{2}$ .
Multiply by Conjugate: Multiply the original expression by a fraction equivalent to $1$ that has the conjugate of the denominator as both its numerator and denominator. $\newline$ To rationalize the denominator, we multiply the original expression by ( $5$ + sqrt( $2$ ))/( $5$ + sqrt( $2$ )). $\newline$ $\frac{5}{5 - \sqrt{2}} \times \frac{5 + \sqrt{2}}{5 + \sqrt{2}}$
Multiply in Numerator: Perform the multiplication in the numerator. $\newline$ $5 \times (5 + \sqrt{2}) = 5 \times 5 + 5 \times \sqrt{2} = 25 + 5\sqrt{2}$
Multiply in Denominator: Perform the multiplication in the denominator using the difference of squares formula. $\newline$ $(5 - \sqrt{2}) \times (5 + \sqrt{2}) = 5^2 - (\sqrt{2})^2 = 25 - 2 = 23$
Write Simplified Expression: Write the simplified expression with the rationalized denominator. $\newline$ $\frac{25 + 5\sqrt{2}}{23}$ $\newline$ This fraction is already in simplest form.