36r4 + 36r3 + 3r2) / 9r3

this is normally written as (36r^4 + 36r^3 + 3r^2) / (9r^3)

this is equivalent to (36r^4 / 9r^3) + (36r^3 / 9r^3) + (3r^2 / 9r^3)

the rule for exponent arithmetic to apply is that x^a/x^b = x^(a-b).

so, basically, if the base is the same, then you subtract the exponent in the denominator from the exponent in the numerator and put the result in the numerator.

the other rule for exponent arithmetic to apply is that x^-a = 1/x^a.

you take the reciprocal and change the sign of the exponent.

x^-a = 1/x^a

x^a = 1/x^-a

normally you want to simplify by making all the exponents positive, so if you have x^-a, then show it as 1/x^a, and if you have 1/x^-a, then show it as x^a.

looking at each of the parts of (36r^4 / 9r^3) + (36r^3 / 9r^3) + (3r^2 / 9r^3) individually, you get:

36r^4 / 9r^3 becomes 36/9 * r^4/r^3 which becomes 4 * r^(4-1) which becomes 4r^1 which becomes 4r.

36r^3 / 9r^3 becomes 36/9 * r^3/r^3 which becomes 4 * r^(3-3) which becomes 4 * r^0 which becomes 4 * 1 which becomes 4.

3r^2/9r^3 becomes 3/9 * r^2/r^3 which becomes 1/3 * r^(2-3) which becomes 1/3 * r^-1 which becomes 1/3 * 1/r^1 which becomes 1/3 * 1/r which becomes 1/(3r)

your simplified expression becomes 4r + 4 + 1/(3r)

this should be your solution, but if you wanted to put everything under the same denominator, then you would get:

4r + 4 + 1/(3r) becomes (4r*3r + 4*3r + 1) / (3r)

simplify this to get:

(12r^2 + 12r + 1) / (3r)

that should be your simplified expression.

if we did this correctly, your original expression and your simplified expression should give you the same answer if you replace r with a random value.

for example, assume r = 7

your original expression of (36r^4 + 36r^3 + 3r^2) / (9r^3) becomes equal to 32.04761905

your simplified expression of (12r^2 + 12r + 1) / (3r) becomes equal to 32.04761905

since they both provide the same answer, they're equivalent and you can reasonable conclude that you simplified correctly.

if your stated solution is not this, then let me know what it is and i'll come up with a reason why it should be that and not what i just told you.

the difference, if any, is usually in what the definition of simplified is.