a+h a
_____ -- ______
a+h+1 a+1
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h
To simplify the expression (a+h)/(a+h+1) - a/(a+1), we need to create a common denominator for the fractions.
The common denominator will be (a+h+1) * (a+1).
Now, let's find the numerators:
For the first fraction, (a+h)/(a+h+1), the numerator is (a+h).
For the second fraction, a/(a+1), the numerator is a.
So, rewriting the expression with the common denominator, we have:
[(a+h)*(a+1)]/[(a+h+1)*(a+1)] - a*(a+h+1)/[(a+1)*(a+h+1)].
Now, we can combine the fractions and simplify further:
[(a^2 + ah + a + h) - a^2 - ah - a]/[(a+1)*(a+h+1)].
Simplifying further:
[(a^2 + ah + a + h - a^2 - ah - a)]/[(a+1)*(a+h+1)].
Now, let's simplify the numerator:
(a^2 - a^2) + (ah - ah) + (a - a) + h]/[(a+1)*(a+h+1)].
After cancelling out like terms, we are left with just h in the numerator:
h/[(a+1)*(a+h+1)].
So, the simplified expression is h/[(a+1)*(a+h+1)].