To simplify the expression \((1 + \frac{3}{4})^2 : (2 + \frac{1}{3})^3\), follow these steps:

### Step 1: Simplify each term inside the parentheses
1. \(1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}\)
2. \(2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)

### Step 2: Raise the simplified terms to their respective powers
1. \(\left(\frac{7}{4}\right)^2 = \frac{7^2}{4^2} = \frac{49}{16}\)
2. \(\left(\frac{7}{3}\right)^3 = \frac{7^3}{3^3} = \frac{343}{27}\)

### Step 3: Divide the two results
\[
\frac{\left(\frac{49}{16}\right)}{\left(\frac{343}{27}\right)} = \frac{49}{16} \times \frac{27}{343}
\]

### Step 4: Simplify the multiplication
1. Multiply the numerators: \(49 \times 27 = 1323\)
2. Multiply the denominators: \(16 \times 343 = 5488\)

Thus, the simplified result is:
\[
\frac{1323}{5488}
\]

This fraction can no longer be simplified, so the final answer is \(\frac{1323}{5488}\).

Would you like further details on this process or have any questions?

### Related questions:
1. How do you simplify more complex fractional exponents?
2. What is the process of dividing fractions in general?
3. Can this expression be written as a decimal approximation?
4. How do you simplify roots and powers with fractions?
5. How do fractional powers relate to radicals?

### Tip:
When dividing fractions, remember to multiply by the reciprocal of the divisor to make the calculation easier!

Explore the step-by-step process to simplify the expression (1 + 3/4)^2 ÷ (2 + 1/3)^3. This problem covers essential concepts in fractions, exponents, and division of fractions, making it perfect for students in Grades 6-8.

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