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Calculate this with every step shown and why

To solve \(\left(1 \frac{3}{2}\right)^2\) step-by-step:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \(1 \frac{3}{2}\) can be converted to an improper fraction as follows:
\[
1 \frac{3}{2} = \frac{2 \times 1 + 3}{2} = \frac{5}{2}
\]

### Step 2: Write the problem using the improper fraction
Substituting \(\frac{5}{2}\) for \(1 \frac{3}{2}\), we now have:
\[
\left(\frac{5}{2}\right)^2
\]

### Step 3: Square the fraction
When squaring a fraction, square the numerator and the denominator separately:
\[
\left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4}
\]

### Step 4: Simplify if necessary
The fraction \(\frac{25}{4}\) is already in its simplest form, but if needed, you can express it as a mixed number:
\[
\frac{25}{4} = 6 \frac{1}{4}
\]

### Final Answer:
\[
\left(1 \frac{3}{2}\right)^2 = \frac{25}{4} \quad \text{or} \quad 6 \frac{1}{4}
\]

### Why each step matters:
1. **Converting to an improper fraction** ensures the calculation can proceed algebraically.
2. **Squaring each part of the fraction separately** respects the rules of exponents.
3. **Simplifying the result** makes it more interpretable.

Would you like further clarification or a visual explanation? Here are five related questions to expand your understanding:

1. How do you convert any mixed number into an improper fraction?
2. What is the general rule for squaring fractions?
3. Can this problem be solved using decimal representation instead of fractions?
4. Why is it useful to simplify improper fractions to mixed numbers?
5. How does squaring negative fractions differ from squaring positive fractions?

**Tip:** Always convert mixed numbers to improper fractions before performing operations like squaring or multiplying for accuracy and consistency.

Understand how to square the mixed number (1 3/2)^2 with a detailed, step-by-step explanation. This guide covers converting mixed numbers to improper fractions, applying exponent rules, and simplifying the result. Ideal for students in Grades 6-8.

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