Simplify. Rationalize the denominator. 7/-6 + sqrt 5 ______

Find Conjugate: Select the conjugate of -6 + sqrt 5. Conjugate of a number in the form a + sqrt(b) is a - sqrt(b). Therefore, the conjugate of -6 + sqrt 5 is -6 - 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 (-6 - sqrt 5)/(-6 - sqrt 5). This gives us (7 * (-6 - sqrt 5))/((-6 + sqrt 5) * (-6 - sqrt 5)).Simplify Numerator: Simplify the numerator. Multiplying 7 by each term in the conjugate -6 - sqrt 5 gives us: 7 * (-6) - 7 * sqrt(5) = -42 - 7 * sqrt(5).Simplify Denominator: Simplify the denominator using the difference of squares formula. The denominator is in the form (a + b)(a - b), which simplifies to a^2 - b^2. So, (-6 + sqrt 5) * (-6 - sqrt 5) simplifies to: (-6)^2 - (sqrt(5))^2 = 36 - 5 = 31.Write Simplified Fraction: Write the simplified fraction. The fraction now is (-42 - 7 * sqrt(5))/31. This fraction is already in simplest form.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Simplify. Rationalize the denominator. $\newline$ $\frac{7}{-6 + \sqrt{5}}$

View step-by-step help

Home

Math Problems

Algebra 2

Simplify radical expressions using conjugates

Full solution

Q. Simplify. Rationalize the denominator.

\newline

\frac{7}{-6 + \sqrt{5}}

Find Conjugate: Select the conjugate of $-6 + \sqrt{5}$ . $\newline$ Conjugate of a number in the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . $\newline$ Therefore, the conjugate of $-6 + \sqrt{5}$ is $-6 - \sqrt{5}$ .
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 $(-6 - \sqrt{5})/(-6 - \sqrt{5})$ . $\newline$ This gives us $(7 \cdot (-6 - \sqrt{5}))/((-6 + \sqrt{5}) \cdot (-6 - \sqrt{5}))$ .
Simplify Numerator: Simplify the numerator. $\newline$ Multiplying $7$ by each term in the conjugate $-6 - \sqrt{5}$ gives us: $\newline$ $7 \times (-6) - 7 \times \sqrt{5}$ $\newline$ $= -42 - 7 \times \sqrt{5}$ .
Simplify Denominator: Simplify the denominator using the difference of squares formula. $\newline$ The denominator is in the form $(a + b)(a - b)$ , which simplifies to $a^2 - b^2$ . $\newline$ So, $(-6 + \sqrt{5}) * (-6 - \sqrt{5})$ simplifies to: $\newline$ $(-6)^2 - (\sqrt{5})^2$ $\newline$ $= 36 - 5$ $\newline$ $= 31$ .
Write Simplified Fraction: Write the simplified fraction. $\newline$ The fraction now is $(-42 - 7 \sqrt{5})/31$ . $\newline$ This fraction is already in simplest form.