Simplify. Rationalize the denominator. 8/4 - sqrt 5 ______

Identify Conjugate of Denominator: Identify the conjugate of the denominator. The conjugate of a - sqrt(b) is a + sqrt(b). Therefore, the conjugate of 4 - sqrt(5) is 4 + 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 (4 + sqrt(5))/(4 + sqrt(5)). (8/(4 - sqrt(5))) * ((4 + sqrt(5))/(4 + sqrt(5)))Apply Distributive Property: Apply the distributive property to multiply the numerators. 8 * (4 + sqrt(5)) = 8*4 + 8*sqrt(5) = 32 + 8*sqrt(5)Apply Difference of Squares: Apply the difference of squares formula to multiply the denominators. (4 - sqrt(5)) * (4 + sqrt(5)) = 4^2 - (sqrt(5))^2 = 16 - 5 = 11Write Simplified Expression: Write the simplified expression with a rationalized denominator. (32 + 8*sqrt(5)) / 11

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Simplify. Rationalize the denominator. $\newline$ $\frac{8}{4 - \sqrt{5}}$

View step-by-step help

Home

Math Problems

Algebra 1

Simplify radical expressions using conjugates

Full solution

Q. Simplify. Rationalize the denominator.

\newline

\frac{8}{4 - \sqrt{5}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$ . Therefore, the conjugate of $4 - \sqrt{5}$ is $4 + \sqrt{5}$ .
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 $(4 + \sqrt{5})/(4 + \sqrt{5})$ . $\newline$ $(8/(4 - \sqrt{5})) \cdot ((4 + \sqrt{5})/(4 + \sqrt{5}))$
Apply Distributive Property: Apply the distributive property to multiply the numerators. $\newline$ $8 \times (4 + \sqrt{5}) = 8\times4 + 8\times\sqrt{5} = 32 + 8\times\sqrt{5}$
Apply Difference of Squares: Apply the difference of squares formula to multiply the denominators. $\newline$ $(4 - \sqrt{5}) \times (4 + \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11$
Write Simplified Expression: Write the simplified expression with a rationalized denominator. $\newline$ $(\frac{32 + 8\sqrt{5}}{11})$