What is the common denominator?
3 2
- + -
1 a-1
-------
4 3
- - -
1 a+1
If your denominator on the top is (a-1) and your denominator on the bottom is (a+1) then your common denominator is (a-1)(a+1) which by the way is (a^2-1)
Thank you very much. It helped me not only on that problem but others as well. :)
"........but others as well."
Good! That is the whole idea :) You are very welcome.
To find the common denominator, you need to determine the least common multiple (LCM) of the denominators of the fractions involved.
In this case, the denominators of the fractions are 1, a-1, 4, and a+1. To find the LCM of these denominators, you can list out their multiples and identify the smallest number that appears in all the lists.
Let's start by finding the multiples of each denominator:
- Multiples of 1: 1, 2, 3, 4, ...
- Multiples of a-1: (a-1), 2(a-1), 3(a-1), 4(a-1), ...
- Multiples of 4: 4, 8, 12, 16, ...
- Multiples of a+1: (a+1), 2(a+1), 3(a+1), 4(a+1), ...
Looking at the lists, it's clear that the smallest number that appears in all four lists is 4(a-1)(a+1). Therefore, the common denominator is 4(a-1)(a+1).
Now, let's rewrite the fractions with this common denominator:
3/1 = (3/1) * ((a-1)(a+1))/((a-1)(a+1))
2/(a-1) = (2/(a-1)) * (4/4) = (8/(4(a-1)))
1/4 = (1/4) * ((a+1)(a-1))/((a+1)(a-1))
3/(a+1) = (3/(a+1)) * (4/4) = (12/(4(a+1)))
Substituting these equivalent fractions into the original expression, we have:
(3/1) * ((a-1)(a+1))/((a-1)(a+1)) + (8/(4(a-1))) - (1/4) * ((a+1)(a-1))/((a+1)(a-1)) - (12/(4(a+1)))
Now, you can simplify the expression further or perform any necessary operations.