Math Problem Statement

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

Solution

The expression in the image asks to simplify:

349÷(51234)+59103 \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:

    • 3493 \frac{4}{9} becomes 319\frac{31}{9}
    • 59105 \frac{9}{10} becomes 5910\frac{59}{10}
  2. Simplify the exponent 2342^{\frac{3}{4}}: 2342^{\frac{3}{4}} represents the 4th root of 2 cubed. You can approximate this value using standard methods: 2341.681792^{\frac{3}{4}} \approx 1.68179

  3. Simplify the expression inside the parentheses: 51234=51.681793.318215^1 - 2^{\frac{3}{4}} = 5 - 1.68179 \approx 3.31821

  4. Divide 319÷3.31821\frac{31}{9} \div 3.31821: 319÷3.31821319×13.318211.0379\frac{31}{9} \div 3.31821 \approx \frac{31}{9} \times \frac{1}{3.31821} \approx 1.0379

  5. Add the result to 59105 \frac{9}{10} (or 5910\frac{59}{10}): 1.0379+59101.0379+5.9=6.93791.0379 + \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 2342^{\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.

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Math Problem Analysis

Mathematical Concepts

Fractions
Exponents
Order of Operations
Division
Addition

Formulas

Mixed fraction to improper fraction conversion: a b/c = (ac + b)/c
Exponent rule: x^(m/n) = n√(x^m)

Theorems

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Suitable Grade Level

Grades 9-11