Let's break down and solve the expression step by step:

Given expression:

\[
\frac{{(6+3)^2 + (9 - 10 \div 5)}}{{10 - 16 \div 8 + 9}}
\]

### Step 1: Simplify the numerator

1. **Simplify inside the parentheses:**
   - \(6 + 3 = 9\)
   - \(9 - 10 \div 5\)

2. **Exponentiation:**
   - \((6 + 3)^2 = 9^2 = 81\)

3. **Division inside the parentheses:**
   - \(10 \div 5 = 2\)

4. **Subtraction inside the parentheses:**
   - \(9 - 2 = 7\)

5. **Result for the numerator:**
   - \(81 + 7 = 88\)

### Step 2: Simplify the denominator

1. **Division inside the denominator:**
   - \(16 \div 8 = 2\)

2. **Simplify inside the denominator:**
   - \(10 - 2 + 9 = 17\)

### Step 3: Combine numerator and denominator

\[
\frac{88}{17}
\]

This simplifies to:

\[
\frac{88}{17} \approx 5.176
\]

So, the result of the expression is approximately \(5.176\).

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Do you want details or have any questions? 

Here are 5 related questions:

1. How does the order of operations (PEMDAS/BODMAS) affect the outcome of such expressions?
2. What are the differences between using parentheses and brackets in mathematical expressions?
3. Can you explain how to simplify expressions with more complex exponents or roots?
4. How would the result change if the operations in the expression were rearranged?
5. What is the significance of simplifying both the numerator and denominator before dividing?

**Tip:** Always remember to follow the order of operations strictly to avoid errors in solving complex expressions!

Learn how to solve the arithmetic expression ((6+3)^2 + (9 - 10 ÷ 5))/(10 - 16 ÷ 8 + 9) step-by-step. This problem covers arithmetic operations, exponents, fractions, and the order of operations (PEMDAS/BODMAS). Suitable for students in Grades 6-8.

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