I need to express the following in equivalent form:

m-2/m^2-5m+6 , 2x+6/x^2-6x-27 , a^2+a/a^2+2a+1 , 2w+1/2w^2+2w+1 , 8Z+6Z^2/9z^2-16

I really don't understand how I can do this if the denominator is smaller and I just don't see how I can simplify since I think that's what it's asking; I really just need an overall explanation how to do it, not the answer to them all;. Thanks.

To express each of the given expressions in an equivalent form, we need to simplify the expressions by factoring, canceling common factors, and simplifying further if possible.

Here is an overall explanation of the steps to simplify rational expressions:

Step 1: Factor the numerator and denominator (if possible).

To factor an algebraic expression, look for common factors, apply the distributive property, or use factoring techniques such as difference of squares, perfect square trinomials, or factoring by grouping.

Step 2: Cancel common factors.

If there are common factors in the numerator and denominator, you can cancel them out to simplify the expression. Be cautious when canceling, as it should only be done if the factors are identical in both the numerator and denominator.

Step 3: Simplify further (if possible).

After canceling the common factors, check if the remaining expressions can be factored or simplified further.

Now let's apply these steps to the given expressions:

1. m - 2 / m^2 - 5m + 6:
Factor the numerator and denominator:
Numerator: m - 2
Denominator: (m - 2)(m - 3)
Cancel out the common factor (m - 2):
Simplified form: 1 / (m - 3)

2. 2x + 6 / x^2 - 6x - 27:
Factor the numerator and denominator:
Numerator: 2(x + 3)
Denominator: (x - 9)(x + 3)
Cancel out the common factor (x + 3):
Simplified form: 2 / (x - 9)

3. a^2 + a / a^2 + 2a + 1:
The numerator cannot be factored further.
Denominator: (a + 1)(a + 1)
Cancel out the common factor (a + 1):
Simplified form: a / (a + 1)

4. 2w + 1 / 2w^2 + 2w + 1:
The numerator cannot be factored further.
Denominator: (2w + 1)(w + 1)
Cancel out the common factor (2w + 1):
Simplified form: 1 / (w + 1)

5. 8Z + 6Z^2 / 9z^2 - 16:
Factor the numerator and denominator:
Numerator: 2Z(4 + 3Z)
Denominator: (3z - 4)(3z + 4)
No common factors to cancel out.
Simplified form: 2Z(4 + 3Z) / ((3z - 4)(3z + 4))

Remember to always check for restrictions where the original expression is undefined. This would include any values that would make the denominator zero, as division by zero is undefined.

I hope this explanation helps you understand how to simplify rational expressions with smaller denominators.