Reduce to lowest terms

3x-6/x^2-4x+4

perform operation
w^2-1/(w-1)2 times w-1/w^2+2w+1

Find LCD and convert rational express. to equivalent rat. exp. with LCD as the denominator:
4/a-6 , 5/6-a

To reduce the rational expression (3x-6)/(x^2-4x+4) to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by that GCD.

1. Factorize the numerator and the denominator if possible:
The numerator, 3x-6, can be factorized as 3(x-2).
The denominator, x^2-4x+4, is a perfect square trinomial and can be factorized as (x-2)^2.

2. Divide both the numerator and the denominator by the GCD, which in this case is 3:
(3x-6)/(x^2-4x+4) = 3(x-2)/[(x-2)(x-2)] = 3/(x-2)

So, the reduced form of the given rational expression is 3/(x-2).

Moving on to the next question:

To perform the operation (w^2-1)/[(w-1)^2] times (w-1)/(w^2+2w+1), we need to multiply the two rational expressions together.

1. First, factorize the numerator and the denominator of each rational expression:
(w^2-1) can be factorized as (w-1)(w+1).
[(w-1)^2] is already in its factored form.
(w^2+2w+1) can be factorized as (w+1)(w+1), or (w+1)^2.

2. Now, multiply the two rational expressions:
[(w^2-1)/[(w-1)^2]] * [(w-1)/(w^2+2w+1)] = [(w-1)(w+1)/[(w-1)(w-1)]] * [(w-1)/[(w+1)(w+1)]]

3. Simplify by canceling common factors:
(w-1) in the numerator and the denominator cancel:
[(w+1)/[(w-1)]] * [1/[(w+1)(w+1)]]

So, the final form of the given expression after performing the operation is 1/[(w+1)(w+1)].

Now let's move on to the last question:

To find the least common denominator (LCD) and convert the rational expressions to equivalent rational expressions with the LCD as the denominator, we follow these steps:

1. Identify the denominators of the given rational expressions. In this case, the denominators are (a-6) and (6-a).

2. To find the LCD, we need to factorize the denominators. Since (6-a) is the same as -(a-6), we can write it as -(a-6).
The first denominator is (a-6).
The second denominator is -(a-6).

3. The LCD is the common multiple of both denominators. In this case, the common multiple is (a-6)*-(a-6) = (6-a)(a-6).

4. To convert each rational expression to an equivalent rational expression with the LCD as the denominator, we multiply both the numerator and the denominator of each expression by the necessary factor to obtain the LCD.

For the first rational expression, 4/(a-6), we multiply the numerator and the denominator by -(a-6):
[4 * -(a-6)]/[(a-6) * -(a-6)] = (-4a+24)/((6-a)(a-6))

For the second rational expression, 5/(6-a), we multiply the numerator and the denominator by (a-6):
[5 * (a-6)]/[(6-a) * (a-6)] = (5a-30)/((6-a)(a-6))

Therefore, the given rational expressions, 4/(a-6) and 5/(6-a), when converted to equivalent rational expressions with the LCD as the denominator, become (-4a+24)/((6-a)(a-6)) and (5a-30)/((6-a)(a-6)), respectively.