Simplify the quotient 8a²b divided by 4ab².

Step-by-step solution

1. 1

   Split the quotient into independent pieces. A single fraction of monomials can be handled one symbol at a time:

   $\frac{8a^2b}{4ab^2}=\frac{8}{4}\cdot\frac{a^2}{a}\cdot\frac{b}{b^2}$

   This works because multiplication is commutative, and it keeps the exponent bookkeeping separate from the arithmetic.

2. 2

   Reduce the coefficients.

   $\frac84=2$

3. 3

   Apply the quotient rule to $a$. Subtract exponents, top minus bottom:

   $\frac{a^2}{a}=a^{2-1}=a$

4. 4

   Apply the quotient rule to $b$. Here the larger power is on the bottom:

   $\frac{b}{b^2}=b^{1-2}=b^{-1}=\frac1b$

   A negative exponent is the signal that the factor belongs in the denominator — writing the answer as $2ab^{-1}$ is correct but usually not the requested form.

5. 5

   Assemble and check. Multiplying the three results,

   $\frac{8a^2b}{4ab^2}=\frac{2a}{b}$

   valid for $a\neq 0$ and $b\neq 0$ (the original is undefined there). Test with $a=3$, $b=2$: the original is $\frac{8\cdot 9\cdot 2}{4\cdot 3\cdot 4}=\frac{144}{48}=3$, and $\frac{2\cdot 3}{2}=3$ ✓.