Simplify. Rationalize the denominator. 4/-5 + sqrt 2 ______

Identify Conjugate: Select the conjugate of -5 + sqrt(2). 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. (4 * (-5 - sqrt(2)))/((-5 + sqrt(2)) * (-5 - sqrt(2)))Simplify Numerator: Simplify the numerator. Now we distribute the 4 across the terms in the conjugate. 4 * (-5) + 4 * (-sqrt(2)) = -20 - 4sqrt(2)Simplify Denominator: Simplify the denominator. We 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 Simplified Expression: Write the simplified expression. Now we have the simplified numerator and denominator. (-20 - 4sqrt(2)) / 23 This fraction is already in simplest form.

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Simplify. Rationalize the denominator. $\newline$ $\frac{4}{-5 + \sqrt{2}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{4}{-5 + \sqrt{2}}

Identify Conjugate: Select the conjugate of $-5 + \sqrt{2}$ . $\newline$ 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. $\newline$ $(4 \cdot (-5 - \sqrt{2}))/((-5 + \sqrt{2}) \cdot (-5 - \sqrt{2}))$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $4$ across the terms in the conjugate. $\newline$ $4 \times (-5) + 4 \times (-\sqrt{2})$ $\newline$ $= -20 - 4\sqrt{2}$
Simplify Denominator: Simplify the denominator. $\newline$ We 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 Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and denominator. $\newline$ $(-20 - 4\sqrt{2}) / 23$ $\newline$ This fraction is already in simplest form.