How do you rationalize the denominator and simplify #(3-sqrt5)/(4+sqrt5)#?

1 Answer
Johnson Z.
Mar 17, 2016

#(17-7sqrt(5))/11#

Explanation:

#1#. Start by multiplying the numerator and denominator by the conjugate of the fraction's denominator, #4-sqrt(5)#.

#(3-sqrt(5))/(4+sqrt(5))#

#=(3-sqrt(5))/(4+sqrt(5))((4-sqrt(5))/(4-sqrt(5)))#

#2#. Simplify the numerator.

#=(12-3sqrt(5)-4sqrt(5)+5)/(4+sqrt(5))(1/(4-sqrt(5)))#

#=(17-7sqrt(5))/(4+sqrt(5))(1/(4-sqrt(5)))#

#3#. Simplify the denominator. Note that it contains a difference of squares #(color(red)(a^2-b^2=(a+b)(a-b)))#.

#=(17-7sqrt(5))/(16-5)#

#=color(green)(|bar(ul(color(white)(a/a)(17-7sqrt(5))/(11)color(white)(a/a)|)))#

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