How do you rationalize the denominator and simplify #2/(1-sqrt5)#?

2 Answers
Rhys
Apr 28, 2018

#- (1+sqrt(5) )/2 #

Explanation:

To do this we must multiply both numerator and denominator by the denominators conjugate:

#2/(1-sqrt(5) ) xx (1+sqrt(5))/(1+sqrt(5)) #

As this is just the same as multiplying by 1:

Expanding:

#=> (2(1+sqrt(5) )) / (( 1-sqrt(5))(1+sqrt(5)) #

#=> ( 2 + 2sqrt(5) ) / ( 1 - sqrt(5) + sqrt(5) - sqrt(5)sqrt(5) ) #

#=> ( 2 + 2sqrt(5) ) / ( -4 ) #

#=> -1/2 -1/2 sqrt(5) = - (1+sqrt(5) )/2 #

Jim G.
Apr 28, 2018

#-1/2(1+sqrt5)#

Explanation:

#"to "color(blue)"rationalise the denominator"#

#"multiply the numerator/denominator by the "color(blue)"conjugate"#
#"of the denominator"#

#"the conjugate of "1-sqrt5" is "1color(red)(+)sqrt5#

#rArr(2(1+sqrt5))/((1-sqrt5)(1+sqrt5))#

#"expand the denominator using FOIL"#

#(2(1+sqrt5))/(-4)=-1/2(1+sqrt5)#

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