perform the subtraction
(8a+2b)/(8a^(2)+2ab-3b^(2))-(a+2b)/(16a^(2)-ab^(2))
I factored it to ...
2(4a+b)/( (2a-b)(4a+3b) ) - (a+2b)/( a(16a - b^2) )
nothing divides out, no duplication of factors,
you might just as well do
[ (8a+2b)(16a^2 - ab^2) - (a+2b)(8a^2 + 2ab - 3b^2) ] / [(8a^2 + 2ab - 3b^2)(16a^2 - ab^2) ]
expand and hope for the best
I suspect some sort of typing error.
To perform the subtraction of the given expression:
(8a + 2b) / (8a^2 + 2ab - 3b^2) - (a + 2b) / (16a^2 - ab^2)
We need to find a common denominator for both fractions. The common denominator can be found by taking the least common multiple (LCM) of the denominators. First, let's factorize the denominators:
Denominator 1: 8a^2 + 2ab - 3b^2 = (2a - b)(4a + 3b)
Denominator 2: 16a^2 - ab^2 = a(16a - b)(a + b)
To find the LCM, we need to consider the highest power of each factor. The LCM is obtained by multiplying these highest powers together:
LCM = a * (2a - b) * (4a + 3b) * (16a - b) * (a + b)
Now, let's rewrite both fractions with the common denominator:
[(8a + 2b) * a * (16a - b) * (a + b)] / LCM - [(a + 2b) * (2a - b) * (4a + 3b)] / LCM
Next, simplify and expand both numerators:
[8a * a * (16a - b) * (a + b) + 2b * a * (16a - b) * (a + b)] / LCM - [(a * (2a - b) * (4a + 3b) - 2b * (2a - b) * (4a + 3b))] / LCM
Simplify each term in the numerators:
[8a^3 * (16a - b) * (a + b) + 2ab * (16a - b) * (a + b)] / LCM - [2a^2 * (4a + 3b) - 2b^2 * (4a + 3b)] / LCM
Combine like terms within each numerator:
[8a^4 * (16a - b) + 2ab * (16a - b)] / LCM - [2a^2 * (4a + 3b) - 2b^2 * (4a + 3b)] / LCM
Now, subtract the fractions by subtracting the numerators while keeping the common denominator:
[8a^4 * (16a - b) + 2ab * (16a - b) - 2a^2 * (4a + 3b) + 2b^2 * (4a + 3b)] / LCM
Finally, combine like terms within the numerator and simplify if necessary. The resulting expression is the subtraction of the given expression.