Simplify fully 2√5 / 2+√5 ?

1 Answer
Jim G.
Nov 19, 2017

#10-4sqrt5#

Explanation:

#(2sqrt5)/(2+sqrt5)#

#"to simplify we require the denominator to be rational"#

#"to achieve this multiply the numerator/denominator by the "color(blue)"conjugate"" of "2+sqrt5#

#"the conjugate of " 2+sqrt5" is "2color(red)(-)sqrt5#

#"note that "#

#(2+sqrt5)(2-sqrt5)larrcolor(blue)"expand using FOIL"#

#=4cancel(+2sqrt5)cancel(-2sqrt5)-5=-1larrcolor(blue)"rational"#

#rArr(2sqrt5(2-sqrt5))/((2+sqrt5)(2-sqrt5))#

#=(4sqrt5-10)/(-1)=10-4sqrt5#

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