How do you simplify #\frac { 8- \sqrt { 5} } { 9+ \sqrt { 5} }#?

1 Answer
smendyka
Nov 23, 2017

See a solution process below:

Explanation:

To simplify this expression we must rationalize the denominator, or, in other words, eliminate the radical from the denominator. We can do this by multiplying the expression by the appropriate form of #1#:

#(color(red)(9) - color(red)(sqrt(5)))/(color(red)(9) - color(red)(sqrt(5))) xx (8 - sqrt(5))/(9 + sqrt(5)) =>#

#((color(red)(9) xx 8) - color(red)(9)sqrt(5) - 8color(red)(sqrt(5)) + color(red)(sqrt(5))sqrt(5))/((color(red)(9) xx 9) + color(red)(9)sqrt(5) - 9color(red)(sqrt(5)) - color(red)(sqrt(5))sqrt(5)) =>#

#(72 + (-color(red)(9) - 8)color(red)(sqrt(5)) + 5)/(81 + (color(red)(9) - 9)color(red)(sqrt(5)) - 5) =>#

#(72 + 5 + (-17)color(red)(sqrt(5)))/(81 - 5 + 0) =>#

#(77- 17sqrt(5))/76#

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