Simplify. Rationalize the denominator. 5/-10 + sqrt 2 ______

Identify Conjugate: Identify the conjugate of the denominator -10 + sqrt 2. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of -10 + sqrt 2 is -10 - 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 (-10 - sqrt 2)/(-10 - sqrt 2) which is equal to 1 and thus does not change the value of the expression. (5 * (-10 - sqrt 2)) / ((-10 + sqrt 2) * (-10 - sqrt 2))Simplify Numerator: Simplify the numerator by distributing the multiplication. 5 * (-10) + 5 * (-sqrt 2) = -50 - 5 * sqrt 2Simplify Denominator: Simplify the denominator using the difference of squares formula. (-10)^2 - (sqrt 2)^2 = 100 - 2 = 98Write Simplified Expression: Write the simplified expression with the rationalized denominator. (-50 - 5 * sqrt 2) / 98 This fraction is already in simplest form.

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

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{5}{-10 + \sqrt{2}}

Identify Conjugate: Identify the conjugate of the denominator $-10 + \sqrt{2}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-10 + \sqrt{2}$ is $-10 - \sqrt{2}$ .
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 $(-10 - \sqrt{2})/(-10 - \sqrt{2})$ which is equal to $1$ and thus does not change the value of the expression. $\newline$ $(5 \cdot (-10 - \sqrt{2})) / ((-10 + \sqrt{2}) \cdot (-10 - \sqrt{2}))$
Simplify Numerator: Simplify the numerator by distributing the multiplication. $\newline$ $5 \times (-10) + 5 \times (-\sqrt{2})$ $\newline$ = $-50 - 5 \times \sqrt{2}$
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ $(-10)^2 - (\sqrt{2})^2$ $\newline$ = $100 - 2$ $\newline$ = $98$
Write Simplified Expression: Write the simplified expression with the rationalized denominator. $\newline$ $(-50 - 5 \sqrt{2}) / 98$ $\newline$ This fraction is already in simplest form.