Let’s solve the given expression step by step:

The expression is:

$3^2 + 2^2 \times \left( \frac{3 \times 2^2 \div 3 + 5 \times 2^2}{6} + 1^5 \right)$

Step 1: Simplify exponents

$3^2 = 9, \, 2^2 = 4, \, 1^5 = 1$

Now substitute these values: $9 + 4 \times \left( \frac{3 \times 4 \div 3 + 5 \times 4}{6} + 1 \right)$

Step 2: Simplify terms inside the parentheses

First, simplify $3 \times 4 \div 3$: $3 \times 4 = 12, \quad 12 \div 3 = 4$

Next, simplify $5 \times 4$: $5 \times 4 = 20$

Now the expression becomes: $9 + 4 \times \left( \frac{4 + 20}{6} + 1 \right)$

Step 3: Simplify the fraction

$\frac{4 + 20}{6} = \frac{24}{6} = 4$

Substitute back: $9 + 4 \times \left(4 + 1\right)$

Step 4: Simplify the addition inside parentheses

$4 + 1 = 5$

Now the expression is: $9 + 4 \times 5$

Step 5: Simplify the multiplication

$4 \times 5 = 20$

Now the expression becomes: $9 + 20$

Step 6: Final addition

$9 + 20 = 29$

Final Answer:

$\boxed{29}$

Let me know if you'd like further explanation or have additional questions.

Related Questions:

1. How do you prioritize operations in such expressions (PEMDAS)?
2. What are some common mistakes when simplifying expressions with parentheses?
3. How do fractional terms influence multi-step operations like these?
4. What are the applications of these arithmetic calculations in real-world problems?
5. How can exponents affect the growth rate of values compared to multiplication?

Tip:

Always simplify expressions inside parentheses and handle exponents before moving to multiplication, division, addition, or subtraction.