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

Identify Conjugate of Denominator: Identify the conjugate of the denominator -10 + sqrt 5. The conjugate of a + sqrt(b) is a - sqrt(b), so the conjugate of -10 + sqrt 5 is -10 - sqrt 5.Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. To rationalize the denominator, we multiply the fraction by (-10 - sqrt 5)/(-10 - sqrt 5): (2/(-10 + sqrt 5)) * ((-10 - sqrt 5)/(-10 - sqrt 5))Distribute Multiplication in Numerator: Distribute the multiplication in the numerator. Multiply 2 by each term in the conjugate: 2 * (-10) = -20 2 * (-sqrt 5) = -2sqrt 5 So the numerator becomes -20 - 2sqrt 5.Apply Difference of Squares: Apply the difference of squares in the denominator. The product of a binomial and its conjugate is the difference of squares: (-10 + sqrt 5) * (-10 - sqrt 5) = (-10)^2 - (sqrt 5)^2 (-10)^2 = 100 (sqrt 5)^2 = 5 So the denominator becomes 100 - 5.Simplify Denominator: Simplify the denominator. Subtract 5 from 100: 100 - 5 = 95 So the denominator simplifies to 95.Write Simplified Expression: Write the simplified expression. The fraction with the rationalized denominator is: (-20 - 2sqrt 5)/95

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

View step-by-step help

Home

Math Problems

Algebra 2

Simplify radical expressions using conjugates

Full solution

Q. Simplify. Rationalize the denominator.

\newline

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

Identify Conjugate of Denominator: Identify the conjugate of the denominator $-10 + \sqrt{5}$ . $\newline$ The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$ , so the conjugate of $-10 + \sqrt{5}$ is $-10 - \sqrt{5}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the fraction by $(-10 - \sqrt{5})/(-10 - \sqrt{5})$ : $\newline$ $(2/(-10 + \sqrt{5})) \times ((-10 - \sqrt{5})/(-10 - \sqrt{5}))$
Distribute Multiplication in Numerator: Distribute the multiplication in the numerator. $\newline$ Multiply $2$ by each term in the conjugate: $\newline$ $2 \times (-10) = -20$ $\newline$ $2 \times (-\sqrt{5}) = -2\sqrt{5}$ $\newline$ So the numerator becomes $-20 - 2\sqrt{5}$ .
Apply Difference of Squares: Apply the difference of squares in the denominator. $\newline$ The product of a binomial and its conjugate is the difference of squares: $\newline$ $(-10 + \sqrt{5}) * (-10 - \sqrt{5}) = (-10)^2 - (\sqrt{5})^2$ $\newline$ $(-10)^2 = 100$ $\newline$ $(\sqrt{5})^2 = 5$ $\newline$ So the denominator becomes $100 - 5$ .
Simplify Denominator: Simplify the denominator. $\newline$ Subtract $5$ from $100$ : $\newline$ $100 - 5 = 95$ $\newline$ So the denominator simplifies to $95$ .
Write Simplified Expression: Write the simplified expression. $\newline$ The fraction with the rationalized denominator is: $\newline$ $(-20 - 2\sqrt{5})/95$