Simplifying Rational Expressions:

a^3-2a^2/2a^2-4a

a^3-2a^2/2a^2-4a

a^2(a-2)/2a(a-2)
Reduces to
a/2

To simplify the rational expression (a^3 - 2a^2) / (2a^2 - 4a), we can factor out common terms from the numerator and the denominator.

Step 1: Factor out a common term from the numerator and the denominator.
(a^3 - 2a^2) / (2a^2 - 4a) = a^2(a - 2) / 2a(a - 2)

Step 2: Simplify the expression further by canceling out common factors.
Since we have (a - 2) in both the numerator and the denominator, we can cancel them out.
(a^2(a - 2)) / (2a(a - 2)) = a^2 / 2a

Step 3: Simplify the expression by dividing the common factors.
a^2 / 2a = a / 2

Therefore, the simplified form of the rational expression (a^3 - 2a^2) / (2a^2 - 4a) is a / 2.

To simplify the rational expression (a^3 - 2a^2) / (2a^2 - 4a), we can start by factoring both the numerator (a^3 - 2a^2) and the denominator (2a^2 - 4a).

1. Factoring the numerator:
We can factor out a common factor of a^2 from both terms in the numerator:
a^3 - 2a^2 = a^2(a - 2a) = a^2(a^2 - 2)

2. Factoring the denominator:
We can factor out a common factor of 2a from both terms in the denominator:
2a^2 - 4a = 2a(a - 2)

Now, we have factored forms for both the numerator and the denominator:
(a^3 - 2a^2) / (2a^2 - 4a) = (a^2(a^2 - 2)) / (2a(a - 2))

Next, we can look for any common factors between the numerator and the denominator that can be canceled out. In this case, we have a common factor of (a - 2).

3. Canceling out common factors:
We can simplify the expression by canceling out the common factor of (a - 2):
(a^2(a^2 - 2)) / (2a(a - 2))
= (a^2 * 1) / (2a * 1)
= a^2 / (2a)

Thus, the simplified form of the rational expression (a^3 - 2a^2) / (2a^2 - 4a) is a^2 / (2a).