Simplify. Rationalize the denominator. 2/-9 - sqrt 5 ______

Select conjugate: Select the conjugate of -9 - sqrt(5). The conjugate of a number of the form a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of -9 - sqrt(5) is -9 + sqrt(5).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. (2 * (-9 + sqrt(5))) / ((-9 - sqrt(5)) * (-9 + sqrt(5)))Simplify numerator: Simplify the numerator. Now we distribute the 2 in the numerator across the conjugate. 2 * (-9) + 2 * sqrt(5) = -18 + 2sqrt(5)Simplify denominator: Simplify the denominator using the difference of squares formula. The denominator is in the form of (a - b)(a + b), which simplifies to a^2 - b^2. (-9)^2 - (sqrt(5))^2 = 81 - 5 = 76Write simplified expression: Write the simplified expression. Now we have the numerator and the denominator simplified, so we write the expression as: (-18 + 2sqrt(5)) / 76Simplify fraction: Simplify the fraction by dividing both terms in the numerator by the denominator. -18 / 76 + (2sqrt(5)) / 76 This simplifies to: -9 / 38 + sqrt(5) / 38

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Simplify. Rationalize the denominator. $\newline$ $\frac{2}{-9 - \sqrt{5}}$

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}{-9 - \sqrt{5}}

Select conjugate: Select the conjugate of $-9 - \sqrt{5}$ . The conjugate of a number of the form $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $-9 - \sqrt{5}$ is $-9 + \sqrt{5}$ .
Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ $(2 \times (-9 + \sqrt{5})) / ((-9 - \sqrt{5}) \times (-9 + \sqrt{5}))$
Simplify numerator: Simplify the numerator. $\newline$ Now we distribute the $2$ in the numerator across the conjugate. $\newline$ $2 \times (-9) + 2 \times \sqrt{5} = -18 + 2\sqrt{5}$
Simplify denominator: Simplify the denominator using the difference of squares formula. $\newline$ The denominator is in the form of $(a - b)(a + b)$ , which simplifies to $a^2 - b^2$ . $\newline$ $(-9)^2 - (\sqrt{5})^2 = 81 - 5 = 76$
Write simplified expression: Write the simplified expression. $\newline$ Now we have the numerator and the denominator simplified, so we write the expression as: $\newline$ $(-18 + 2\sqrt{5}) / 76$
Simplify fraction: Simplify the fraction by dividing both terms in the numerator by the denominator. $\newline$ $-\frac{18}{76} + \frac{2\sqrt{5}}{76}$ $\newline$ This simplifies to: $\newline$ $-\frac{9}{38} + \frac{\sqrt{5}}{38}$