Carry out the following divisions: I. 28x^4 ÷ 56x II. - 36y^3 ÷ 9y^2 III. 66pq^2r^3 ÷ 11qr^2 IV. 34x^3y^3z^3 ÷ 51xy^2z^3 V. 12a^8b^8 ÷ (- 6a^6b^4)

Let's solve each division step by step.

I. 28x⁴ ÷ 56x
We can divide the coefficients and subtract the exponents of the same base terms:

Coefficients: 28 ÷ 56 = 1/2
For x⁴ ÷ x, subtract the exponents: 4 - 1 = 3
So the result is:
(1/2)x³

II. -36y³ ÷ 9y²
Again, divide the coefficients and subtract the exponents of the same base terms:

Coefficients: -36 ÷ 9 = -4
For y³ ÷ y², subtract the exponents: 3 - 2 = 1
So the result is:
-4y

III. 66pq²r³ ÷ 11qr²
Divide the coefficients and subtract the exponents of the same base terms:

Coefficients: 66 ÷ 11 = 6
For p: p ÷ (no p term in the denominator) = p
For q² ÷ q, subtract the exponents: 2 - 1 = 1
For r³ ÷ r², subtract the exponents: 3 - 2 = 1
So the result is:
6pqr

IV. 34x³y³z³ ÷ 51xy²z³
Divide the coefficients and subtract the exponents of the same base terms:

Coefficients: 34 ÷ 51 = 2/3
For x³ ÷ x, subtract the exponents: 3 - 1 = 2
For y³ ÷ y², subtract the exponents: 3 - 2 = 1
For z³ ÷ z³, subtract the exponents: 3 - 3 = 0 (so z is cancelled out)
So the result is:
(2/3)x²y

V. 12a⁸b⁸ ÷ (-6a⁶b⁴)
Divide the coefficients and subtract the exponents of the same base terms:

Coefficients: 12 ÷ (-6) = -2
For a⁸ ÷ a⁶, subtract the exponents: 8 - 6 = 2
For b⁸ ÷ b⁴, subtract the exponents: 8 - 4 = 4
So the result is:
-2a²b⁴

Final answers: I. (1/2)x³
II. -4y
III. 6pqr
IV. (2/3)x²y
V. -2a²b⁴