the transform of t^n
Among the handful of building-block functions whose transforms you simply memorize, the plain powers of t — t, t^2, t^3, and so on — are the most basic after the constant. They show up everywhere because polynomials are everywhere, and their transform follows a clean, rememberable pattern.
The rule is L{t^n} = n! / s^(n + 1), where n is a non-negative integer and n! (n factorial) means n times (n - 1) times ... times 1. Watch the pattern grow: L{1} = 1/s (taking n = 0, since 0! = 1); L{t} = 1/s^2; L{t^2} = 2/s^3; L{t^3} = 6/s^4. Each higher power of t pushes the power of s in the denominator up by one and multiplies the top by the next factorial. You can derive it by integration by parts, but in practice you read it straight off the table. (For non-integer powers the factorial is replaced by the gamma function, but the integer case is the workhorse.)
This entry is one of the standard transform-table rows you reach for constantly, especially after a partial-fraction decomposition leaves you with terms like 2/s^3 to invert. Seeing 2/s^3 you immediately read it as t^2; seeing 6/s^4 you read t^3. The factorials are the only thing to keep straight — forgetting the n! and writing L{t^2} = 1/s^3 is the single most common slip.
Find L{4 t^2}. By linearity and the power rule, L{4 t^2} = 4 (2! / s^3) = 4 (2 / s^3) = 8 / s^3. Conversely, to invert 8/s^3 you reverse the steps and recover 4 t^2.
L{t^n} = n!/s^(n+1) — the factorial on top is easy to forget.
The factorial is the trap. L{t^2} = 2/s^3, not 1/s^3. When inverting, match the factorial: 1/s^4 is t^3/6 (since you need 6 on top to make t^3), not t^3.