Simplify. Rationalize the denominator. 10/5 + 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 5 + sqrt(2) is 5 - 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. (10 * (5 - sqrt(2))) / ((5 + sqrt(2)) * (5 - sqrt(2)))Simplify numerator: Simplify the numerator. Multiply 10 by each term in the conjugate. 10 * 5 - 10 * sqrt(2) = 50 - 10 * sqrt(2)Simplify denominator: Simplify the denominator. Use the difference of squares formula, which states that (a + b)(a - b) = a^2 - b^2. (5)^2 - (sqrt(2))^2 = 25 - 2 = 23Write final expression: Write the simplified expression. Now that we have simplified both the numerator and the denominator, we can write the final expression. (50 - 10 * sqrt(2)) / 23 This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{5 + \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{5 + \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 $5 + \sqrt{2}$ is $5 - \sqrt{2}$ .
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. $\newline$ $\frac{10 \times (5 - \sqrt{2})}{(5 + \sqrt{2}) \times (5 - \sqrt{2})}$
Simplify numerator: Simplify the numerator. $\newline$ Multiply $10$ by each term in the conjugate. $\newline$ $10 \times 5 - 10 \times \sqrt{2}$ $\newline$ $= 50 - 10 \times \sqrt{2}$
Simplify denominator: Simplify the denominator. $\newline$ Use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ . $\newline$ $(5)^2 - (\sqrt{2})^2$ $\newline$ = $25$ - $2$ $\newline$ = $23$
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. $\newline$ $(50 - 10 \times \sqrt{2}) / 23$ $\newline$ This fraction is already in simplest form.