Let's break down and simplify the given expression:

\[
\left(2 \frac{1}{4}\right)^{-\frac{1}{2}} \times \left(3 \frac{3}{8}\right)^{\frac{1}{3}}
\]

### Step 1: Convert mixed numbers to improper fractions.
- \( 2 \frac{1}{4} = \frac{9}{4} \)
- \( 3 \frac{3}{8} = \frac{27}{8} \)

So the expression becomes:
\[
\left(\frac{9}{4}\right)^{-\frac{1}{2}} \times \left(\frac{27}{8}\right)^{\frac{1}{3}}
\]

### Step 2: Apply the exponents to each fraction.
For the first term, \(\left(\frac{9}{4}\right)^{-\frac{1}{2}}\):
- Taking the square root of \(\frac{9}{4}\) gives \(\frac{3}{2}\).
- Raising to the power of \(-\frac{1}{2}\) inverts it: \(\left(\frac{3}{2}\right)^{-1} = \frac{2}{3}\).

For the second term, \(\left(\frac{27}{8}\right)^{\frac{1}{3}}\):
- Taking the cube root of \(\frac{27}{8}\) gives \(\frac{3}{2}\).

### Step 3: Multiply the simplified terms.
Now we have:
\[
\frac{2}{3} \times \frac{3}{2}
\]

The multiplication simplifies as:
\[
\frac{2 \times 3}{3 \times 2} = 1
\]

### Final Answer:
The result of the expression is:

\[
1
\]

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

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Here are five related questions you might explore:
1. How do you convert mixed numbers to improper fractions?
2. What is the general rule for simplifying expressions with fractional exponents?
3. How does the negative exponent affect the result of a fraction?
4. How do you multiply and divide fractions?
5. What are some real-world applications of fractional exponents?

**Tip:** Always simplify fractions before applying exponents to make calculations easier!

Learn how to simplify the expression (2 1/4)^(-1/2) x (3 3/8)^(1/3) using fractional exponents and mixed number conversion. This step-by-step solution covers key algebraic concepts, including applying the negative exponent theorem and root rules. Suitable for Grades 8-10 students.

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