Let's solve the expression step by step:

Expression:

$\frac{3(20t - 3t^2) - \left(\frac{10t^2 - t^3}{3}\right)}{3^2}$

Step 1: Simplify $3^2$

We know $3^2 = 9$, so the denominator becomes 9: $\frac{3(20t - 3t^2) - \left(\frac{10t^2 - t^3}{3}\right)}{9}$

Step 2: Distribute $3$ in the first term

Distribute the $3$ inside the parentheses $3(20t - 3t^2)$: $3(20t - 3t^2) = 60t - 9t^2$

Step 3: Simplify the second term $\left(\frac{10t^2 - t^3}{3}\right)$

This term is already divided by 3, so we leave it as: $\frac{10t^2 - t^3}{3}$

Step 4: Combine terms

Now, substitute these into the main expression: $\frac{(60t - 9t^2) - \left(\frac{10t^2 - t^3}{3}\right)}{9}$

To combine these terms, we need a common denominator for the numerator. Multiply $60t - 9t^2$ by 3/3: $\frac{(3(60t - 9t^2) - (10t^2 - t^3))}{27}$

Distribute the 3 in the first part: $\frac{(180t - 27t^2 - (10t^2 - t^3))}{27}$

Step 5: Simplify the numerator

Now distribute the negative sign: $180t - 27t^2 - 10t^2 + t^3$ Combine like terms: $t^3 - 37t^2 + 180t$

Step 6: Final expression

So the expression simplifies to: $\frac{t^3 - 37t^2 + 180t}{27}$

This is the simplified form of the given expression.

Would you like further details or have any questions?

Here are 5 related questions to expand on the topic:

1. How would this expression change if the exponent of $t$ were different?
2. What happens if we substitute a specific value for $t$ in the final expression?
3. How would you factor the cubic term in the simplified expression?
4. How can you use this result to find critical points of a related function?
5. How would this change if the denominator were a different number?

Tip: When simplifying expressions with multiple terms and fractions, look for common denominators early to ease the combination of terms.