https://file.aiquickdraw.com/mathbot/file/17266423836237mbc31sw.jpg

The expression in the image asks to simplify:

\[
3 \frac{4}{9} \div \left(5^{1} - 2^{\frac{3}{4}}\right) + 5 \frac{9}{10}
\]

### Step-by-step simplification:

1. **Convert the mixed fractions to improper fractions**:
   - \(3 \frac{4}{9}\) becomes \(\frac{31}{9}\)
   - \(5 \frac{9}{10}\) becomes \(\frac{59}{10}\)

2. **Simplify the exponent \(2^{\frac{3}{4}}\)**:
   \(2^{\frac{3}{4}} \) represents the 4th root of 2 cubed. You can approximate this value using standard methods:
   \[
   2^{\frac{3}{4}} \approx 1.68179
   \]

3. **Simplify the expression inside the parentheses**:
   \[
   5^1 - 2^{\frac{3}{4}} = 5 - 1.68179 \approx 3.31821
   \]

4. **Divide** \(\frac{31}{9} \div 3.31821\):
   \[
   \frac{31}{9} \div 3.31821 \approx \frac{31}{9} \times \frac{1}{3.31821} \approx 1.0379
   \]

5. **Add the result to \(5 \frac{9}{10}\) (or \(\frac{59}{10}\))**:
   \[
   1.0379 + \frac{59}{10} \approx 1.0379 + 5.9 = 6.9379
   \]

So, the simplified result is approximately **6.9379**.

Would you like more details or have any further questions?

### Here are 5 related questions to expand your understanding:
1. How do you convert mixed numbers to improper fractions?
2. What are the steps involved in simplifying expressions with exponents and roots?
3. How do you divide fractions by decimals?
4. Why is it important to approximate irrational numbers like \(2^{\frac{3}{4}}\)?
5. How can you simplify complex arithmetic expressions more effectively without a calculator?

### Tip:
When simplifying complex expressions, break them down into smaller parts and work through each operation systematically to avoid mistakes.

Without the use of mathematical tables or calculators, simplify 3 4/9 ÷ (5^1 - 2^(3/4)) + 5 9/10.

Learn how to simplify the mathematical expression 3 4/9 ÷ (5^1 - 2^(3/4)) + 5 9/10 without the use of a calculator. This solution covers key algebraic concepts, including fraction operations and exponent rules. Suitable for students in Grades 9-11.

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