How do you simplify #\frac{y}{y-\frac{y}{y+\frac{1}{y}}}#?

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Answer 1 · 2017-11-09T07:32:28

#"The Exp.="(y^2+1)/(y^2-y+1).#

For ease of writing, let, #x=y/(y+1/y),# so that, the given

Expression (Exp.) becomes,

Exp.#=y/(y-x).........(star).#

Now, #x=y/(y+1/y)=y/{(y^2+1)/y}=y^2/(y^2+1).#

#:. y-x=y-y^2/(y^2+1)={y(y^2+1)-y^2}/(y^2+1).#

#rArr y-x={y(y^2+1-y)}/(y^2+1).#

Therefore, the Exp. = #y/(y-x)=y-:1/(y-x),#

#=y-:{y(y^2+1-y)}/(y^2+1),#

#=cancel(y)xx(y^2+1)/{cancel(y)(y^2+1-y)}.#

#rArr" The Exp.="(y^2+1)/(y^2-y+1).#