Simplify. Rationalize the denominator. 10/-3 + sqrt 5 ______

Identify conjugate of denominator: Identify the conjugate of the denominator -3 + sqrt 5. The conjugate of a number of the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of -3 + sqrt 5 is -3 - sqrt 5.Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the expression by (-3 - sqrt 5)/(-3 - sqrt 5). This gives us (10 * (-3 - sqrt 5))/((-3 + sqrt 5) * (-3 - sqrt 5)).Simplify numerator by distributing: Simplify the numerator by distributing the multiplication. Multiplying 10 by each term in the conjugate, we get: 10 * (-3) - 10 * sqrt(5) = -30 - 10sqrt(5)Simplify denominator using formula: Simplify the denominator by using the difference of squares formula. The difference of squares formula states that (a + b)(a - b) = a^2 - b^2. Applying this to our denominator: (-3)^2 - (sqrt(5))^2 = 9 - 5 = 4Write simplified expression: Write the simplified expression. Now we have (-30 - 10sqrt(5)) / 4. To simplify this, we can divide each term in the numerator by the denominator: (-30/4) - (10sqrt(5)/4) = -7.5 - 2.5sqrt(5)

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Simplify. Rationalize the denominator. $\newline$ $\frac{10}{-3 + \sqrt{5}}$

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Algebra 2

Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

\frac{10}{-3 + \sqrt{5}}

Identify conjugate of denominator: Identify the conjugate of the denominator $-3 + \sqrt{5}$ . $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $-3 + \sqrt{5}$ is $-3 - \sqrt{5}$ .
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 $(-3 - \sqrt{5})/(-3 - \sqrt{5})$ . $\newline$ This gives us $(10 \cdot (-3 - \sqrt{5}))/((-3 + \sqrt{5}) \cdot (-3 - \sqrt{5}))$ .
Simplify numerator by distributing: Simplify the numerator by distributing the multiplication. $\newline$ Multiplying $10$ by each term in the conjugate, we get: $\newline$ $10 \times (-3) - 10 \times \sqrt{5}$ $\newline$ $= -30 - 10\sqrt{5}$
Simplify denominator using formula: Simplify the denominator by using the difference of squares formula. $\newline$ The difference of squares formula states that $(a + b)(a - b) = a^2 - b^2$ . Applying this to our denominator: $\newline$ $(-3)^2 - (\sqrt{5})^2$ $\newline$ $= 9 - 5$ $\newline$ $= 4$
Write simplified expression: Write the simplified expression. $\newline$ Now we have $(-30 - 10\sqrt{5}) / 4$ . To simplify this, we can divide each term in the numerator by the denominator: $\newline$ $(-30/4) - (10\sqrt{5}/4)$ $\newline$ = $-7.5 - 2.5\sqrt{5}$