the domain
5a^2/6b^3 divided by a^3/ab-b^2
To divide the expression (5a^2/6b^3) by (a^3/ab - b^2), we can simplify and cancel out common factors.
First, let's simplify the expression (a^3/ab - b^2):
(a^3/ab) - b^2 = a^(3-1) - b^2
= a^2 - b^2
Now, let's divide (5a^2/6b^3) by (a^2 - b^2):
(5a^2/6b^3) / (a^2 - b^2) = (5a^2/6b^3) * (1/(a^2 - b^2))
= (5a^2/6b^3) * (1/[(a+b)(a-b)])
By canceling out the common factor (a+b) in the denominator and numerator, we get:
= (5a/6b^3) * (1/(a-b))
Therefore, the simplified expression is (5a/6b^3) * (1/(a-b)).
The domain of the expression is all values of a and b except when the denominators become zero. So the domain is:
a ≠ 0, b ≠ 0, a ≠ b.
To divide the expression (5a^2/6b^3) / (a^3/ab-b^2), follow these steps:
Step 1: Multiply the numerator and denominator of the first fraction by the reciprocal of the second fraction (flip the second fraction). This is done to convert the division into multiplication, so the expression becomes: (5a^2/6b^3) * ((ab-b^2)/a^3).
Step 2: Simplify the expression by canceling out common factors between the numerator and denominator.
The common factors between the numerator (5a^2) and the denominator (a^3) are a^2. This leaves a remaining factor of 5 in the numerator and a remaining factor of a in the denominator.
The common factors between the numerator (ab-b^2) and the denominator (6b^3) are b^2. This leaves a remaining factor of a in the numerator and a remaining factor of 6b in the denominator.
Step 3: Write the simplified expression. The simplified expression after canceling out common factors is: (5a)/(6b).
Therefore, the final simplified expression is (5a)/(6b).