Let's break down the expression step by step:

The expression is:

$\frac{1}{4} + 2^2 - \frac{1}{3} \div \frac{14}{3}$

Step 1: Calculate $2^2$

$2^2 = 4$

Step 2: Convert the mixed number $4 \frac{2}{3}$ into an improper fraction

$4 \frac{2}{3} = \frac{14}{3}$

Step 3: Perform the division $\frac{1}{3} \div \frac{14}{3}$

$\frac{1}{3} \div \frac{14}{3} = \frac{1}{3} \times \frac{3}{14} = \frac{1}{14}$

Step 4: Put everything together

The expression now becomes: $\frac{1}{4} + 4 - \frac{1}{14}$

Step 5: Find a common denominator to simplify

The least common denominator (LCD) of 4 and 14 is 28. So, rewrite the fractions: $\frac{1}{4} = \frac{7}{28}, \quad \frac{1}{14} = \frac{2}{28}$

Now, the expression is: $\frac{7}{28} + 4 - \frac{2}{28}$

Step 6: Simplify

$\frac{7}{28} - \frac{2}{28} = \frac{5}{28}$

So the final expression becomes: $\frac{5}{28} + 4 = 4 \frac{5}{28}$

Thus, the answer is: $4 \frac{5}{28} \quad \text{or} \quad \frac{117}{28}$

Would you like more details on any step?

Here are 5 related questions:

1. How do you convert between mixed numbers and improper fractions?
2. What are the steps to perform fraction division?
3. How can you find the least common denominator of two fractions?
4. What is the order of operations (PEMDAS) in math?
5. How do you add and subtract fractions?

Tip: Always perform powers and parentheses first in a mathematical expression to ensure correct results.