How do you rationalize the denominator and simplify #h / (sqrtx - sqrt( x+h))#?

1 Answer
George C.
Apr 24, 2016

#h/(sqrt(x)-sqrt(x+h))=-(sqrt(x)+sqrt(x+h))#

Explanation:

The difference of squares identity can be written:

#a^2-b^2=(a-b)(a+b)#

We use this with #a=sqrt(x)# and #b=sqrt(x+h)# later.

#color(white)()#
Multiply numerator and denominator by #(sqrt(x)+sqrt(x+h))#:

#h/(sqrt(x)-sqrt(x+h))#

#=(h(sqrt(x)+sqrt(x+h)))/((sqrt(x)-sqrt(x+h))(sqrt(x)+sqrt(x+h))#

#=(h(sqrt(x)+sqrt(x+h)))/((sqrt(x))^2-(sqrt(x+h))^2)#

#=(h(sqrt(x)+sqrt(x+h)))/(x-(x+h))#

#=(color(red)(cancel(color(black)(h)))(sqrt(x)+sqrt(x+h)))/(-color(red)(cancel(color(black)(h))))#

#=-(sqrt(x)+sqrt(x+h))#

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