Simplify: a+1/a-1 times a^2-1/(a+1)^2
To simplify the expression (a + 1/a - 1) * (a^2 - 1)/(a + 1)^2, we can follow these steps:
Step 1: Simplify the expression within the parentheses (a + 1/a - 1):
This can be simplified by finding a common denominator for the terms (a + 1/a) and subtracting 1:
- The common denominator for (a + 1/a) is a, so we need to multiply the numerator and denominator of 1 by a: (a * a + 1 * a - 1 * a)/(a).
- This becomes (a^2 + a - a)/(a), simplifying to (a^2)/a.
- Simplifying further, (a^2)/a is equal to a.
So, the expression within the parentheses (a + 1/a - 1) simplifies to a.
Step 2: Simplify the expression (a^2 - 1)/(a + 1)^2:
We can factorize the numerator and the denominator to simplify the expression:
- The numerator (a^2 - 1) is a difference of squares and can be factored as (a + 1)(a - 1).
- The denominator (a + 1)^2 can be expanded as (a + 1)(a + 1).
Now, the expression (a^2 - 1)/(a + 1)^2 can be written as (a + 1)(a - 1)/(a + 1)(a + 1).
Step 3: Simplify further by canceling out the common factors:
Since (a + 1) appears in both the numerator and denominator, we can cancel it out:
(a + 1)(a - 1)/(a + 1)(a + 1) = (a - 1)/(a + 1).
So, the simplified expression is (a - 1)/(a + 1).