Powers Before Products

You now have every piece of the order of operations. First anything that groups: parentheses, fraction bars, root signs and absolute value bars. Then powers and roots. Then multiplication and division. Last, addition and subtraction.

So in $2\cdot {3}^{2}$ the power goes first: ${3}^{2}$ is 9, and $2\cdot 9$ is 18. Multiplying first would give ${6}^{2}$, which is 36, a different number. The exponent sits on the 3 alone, and a power is a product already packaged around its base, so it is finished before the base can be used.

Many people remember the order as PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. The word hides a trap. M comes before D in it, but multiplication does not go before division: they share a rank. So do addition and subtraction, and within a rank you work left to right.