How do you rationalize the denominator and simplify 2/(1-sqrt5)21−√5?

2 Answers
Rhys
Apr 28, 2018

- (1+sqrt(5) )/2 −1+√52

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)) 21−√5×1+√51+√5

As this is just the same as multiplying by 1:

Expanding:

=> (2(1+sqrt(5) )) / (( 1-sqrt(5))(1+sqrt(5)) ⇒2(1+√5)(1−√5)(1+√5)

=> ( 2 + 2sqrt(5) ) / ( 1 - sqrt(5) + sqrt(5) - sqrt(5)sqrt(5) ) ⇒2+2√51−√5+√5−√5√5

=> ( 2 + 2sqrt(5) ) / ( -4 ) ⇒2+2√5−4

=> -1/2 -1/2 sqrt(5) = - (1+sqrt(5) )/2 ⇒−12−12√5=−1+√52

Jim G.
Apr 28, 2018

-1/2(1+sqrt5)−12(1+√5)

Explanation:

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

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

"the conjugate of "1-sqrt5" is "1color(red)(+)sqrt5the conjugate of 1−√5 is 1+√5

rArr(2(1+sqrt5))/((1-sqrt5)(1+sqrt5))⇒2(1+√5)(1−√5)(1+√5)

"expand the denominator using FOIL"expand the denominator using FOIL

(2(1+sqrt5))/(-4)=-1/2(1+sqrt5)2(1+√5)−4=−12(1+√5)

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