To compare the two numbers \(2 \frac{1}{4}^2\) and \(1 \frac{1}{2}^3\), let's first convert them into improper fractions and then simplify.

### Step 1: Convert to improper fractions and calculate the powers.

1. \(2 \frac{1}{4}^2\)
   - Convert \(2 \frac{1}{4}\) to an improper fraction:
     \[
     2 \frac{1}{4} = \frac{9}{4}
     \]
   - Then square it:
     \[
     \left(\frac{9}{4}\right)^2 = \frac{81}{16}
     \]

2. \(1 \frac{1}{2}^3\)
   - Convert \(1 \frac{1}{2}\) to an improper fraction:
     \[
     1 \frac{1}{2} = \frac{3}{2}
     \]
   - Then cube it:
     \[
     \left(\frac{3}{2}\right)^3 = \frac{27}{8}
     \]

### Step 2: Compare the results.

To compare \(\frac{81}{16}\) and \(\frac{27}{8}\), convert them to have a common denominator:

- Convert \(\frac{27}{8}\) to have a denominator of 16:
  \[
  \frac{27}{8} = \frac{27 \times 2}{8 \times 2} = \frac{54}{16}
  \]

Now compare \(\frac{81}{16}\) and \(\frac{54}{16}\):
- Since \(81 > 54\), we have:
  \[
  \frac{81}{16} > \frac{54}{16}
  \]

Thus, \(2 \frac{1}{4}^2\) is greater than \(1 \frac{1}{2}^3\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you compare two fractions with different denominators?
2. What is the result of raising a mixed number to a higher power, like 4 or 5?
3. How would you simplify \(3 \frac{2}{5}^2\)?
4. How do you convert an improper fraction back to a mixed number?
5. What are some strategies to compare exponents without calculating them directly?

**Tip:** When comparing powers of fractions, converting them to improper fractions first can simplify the process and make comparisons more straightforward.

Learn how to compare mixed numbers with powers in fractions. Step-by-step solution to comparing 2 1:4^2 and 1 1:2^3, converting to improper fractions, and determining which is larger. Suitable for Grades 5-7.

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