[3(20t - 3t^2) - (10t^2 - t^3)(1/3)] / 3^2

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.

The query submitted by the user: [3(20t - 3t^2) - (10t^2 - t^3)(1/3)] / 3^2

This page provides a detailed step-by-step solution to the algebraic expression [3(20t - 3t^2) - (10t^2 - t^3)/3] / 9. Learn how to simplify polynomials, apply the distributive property, and combine like terms. Suitable for students in Grades 9-12.

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