Where m = 3 and n = 2, evaluate m - m^2n/(3n).

Step-by-step solution

1. 1

   Read the fraction bar as a grouping symbol. The expression is $m$ minus the single fraction $\dfrac{m^2n}{3n}$ — the whole $m^2n$ is on top and the whole $3n$ underneath. Writing it inline as m - m^2n/3n invites the wrong reading $m-\dfrac{m^2n}{3}\cdot n$, which would give a different value, so fix the grouping before substituting.

2. 2

   Substitute both values. With $m=3$ and $n=2$:

   $3-\frac{3^2\cdot 2}{3\cdot 2}$

   Powers come before multiplication, so $3^2=9$ is evaluated first, not $(3\cdot 2)^2$.

3. 3

   Simplify numerator and denominator separately.

   $3-\frac{9\cdot 2}{6}=3-\frac{18}{6}=3-3$

4. 4

   Finish the subtraction.

   $3-3=0$

   So the expression evaluates to $0$ at these values.

5. 5

   Simplify symbolically to see why it is a near-miss. Cancelling the common $n$ (legal because $n=2\neq 0$) gives

   $m-\frac{m^2n}{3n}=m-\frac{m^2}{3}=m\left(1-\frac{m}{3}\right)$

   This is $0$ exactly when $m=0$ or $m=3$, for every nonzero $n$. The answer $0$ therefore comes from $m=3$ alone; the value of $n$ never mattered.