Adding Rational Expressions

9/xy^3+m - 7ap/xy^3m=

x/3+c/y+4/z=

On your first question, is the denominator of the second term

xy^3m or xy^3+m ?
It makes a big difference.

For your second question, form a common denominator, 3xz, and combine the numerators. This results in
(xyz + 3cz + 12y)/3yz

these are the solutions for the 2 questions

I don't know if you are providing solutions or questions. Anyway, my previous comments still apply.

To add rational expressions, you first need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators. Once you have a common denominator, you can add the fractions together by combining the numerators.

Let's solve the first expression:

Given:
9/xy^3+m - 7ap/xy^3m

Step 1: Find the common denominator.
The denominators are xy^3+m and xy^3m. To find the LCM, we need to factorize the denominators into prime factors:
xy^3+m = x * y * y * y * (m + 1)
xy^3m = x * y * y * y * m

The LCM can be found by taking the highest power of each prime factor:
LCM = x * y * y * y * (m + 1)

Step 2: Create equivalent fractions with the common denominator.
To create the equivalent fractions, multiply each fraction by a form of 1 that does not change their value. In this case, we can multiply the first fraction by (xy^3m)/(xy^3m) and the second fraction by (xy^3+m)/(xy^3+m):
9/xy^3+m * (xy^3m/xy^3m) = 9xy^3m/(xy^3+m * xy^3m)
7ap/xy^3m * (xy^3+m/xy^3+m) = 7ap * (xy^3+m)/(xy^3+m * xy^3m)

The fractions become:
9xy^3m/(xy^3+m * xy^3m) - 7ap * (xy^3+m)/(xy^3+m * xy^3m)

Step 3: Combine the numerators.
Now that we have a common denominator, we can subtract the fractions by subtracting the numerators:
(9xy^3m - 7ap * (xy^3+m))/(xy^3+m * xy^3m)

Now, let's solve the second expression:

Given:
x/3+c/y+4/z

Step 1: Find the common denominator.
The denominators are 3, y, and z. There are no common factors, so the common denominator is 3 * y * z.

Step 2: Create equivalent fractions with the common denominator.
To create the equivalent fractions, multiply each fraction by a form of 1 that does not change their value. In this case, we can multiply the first fraction by (yz)/(yz), the second fraction by (3z)/(3z), and the third fraction by (3y)/(3y):
(x/3) * (yz/yz) = xyz/(3 * y * z)
(c/y) * (3z/3z) = (c * 3z)/(3 * y * z)
(4/z) * (3y/3y) = (4 * 3y)/(3 * y * z)

The fractions become:
xyz/(3 * y * z) + (c * 3z)/(3 * y * z) + (4 * 3y)/(3 * y * z)

Step 3: Combine the numerators.
Now that we have a common denominator, we can add the fractions by adding the numerators:
(xyz + c * 3z + 4 * 3y)/(3 * y * z)

So, the simplified expression is (xyz + c * 3z + 4 * 3y)/(3 * y * z) for the second expression.