How do you evaluate #\frac { 6} { 4- 2\sqrt { 5} }#?

1 Answer
smendyka
Jul 5, 2017

See a solution process below:

Explanation:

First, take the square root of #5#:

#6/(4 - (2 xx 2.236)) = 6/(4 - 4.472) = 6/-0.472 =#

#-12.708# rounded to the nearest thousandth.

If what you want is the fraction to be rationalized we would follow this process:

#(4 + 2sqrt(5))/(color(red)(4) + color(red)(2sqrt(5))) xx 6/(color(blue)(4) - color(blue)(2sqrt(5))) =>#

#(6(4 + 2sqrt(5)))/((color(red)(4) * color(blue)(4)) - (color(red)(4) * color(blue)(2sqrt(5))) + (color(blue)(4) * color(red)(2sqrt(5))) - (color(blue)(2sqrt(5)) * color(red)(2sqrt(5))) =>#

#(6(4 + 2sqrt(5)))/(16 - 8sqrt(5) + 8sqrt(5) - ((2 * 2)(sqrt(5) * sqrt(5))) =>#

#(6(4 + 2sqrt(5)))/(16 - 0 - (4 * 5)) =>#

#(6(4 + 2sqrt(5)))/(16 - 20) =>#

#(6(4 + 2sqrt(5)))/-4 =>#

#((6 * 4) + (6 * 2sqrt(5)))/-4 =>#

#(24 + 12sqrt(5))/-4 =>#

#24/-4 + (12sqrt(5))/-4 =>#

#-6 - 3sqrt(5)#

Substituting #-0.472# for #sqrt(5)# gives:

#-6 - (3 * 2.236) =>#

#-6 - 6.708 =>#

#-12.708#

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