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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