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The expression shown is \( 4^{-3/2} \).

We can simplify this step by step:

1. First, recognize that a negative exponent indicates the reciprocal, so:
   \[
   4^{-3/2} = \frac{1}{4^{3/2}}
   \]

2. Now, express \( 4^{3/2} \) as a power:
   \[
   4^{3/2} = (4^{1/2})^3 = ( \sqrt{4})^3
   \]

3. Since \( \sqrt{4} = 2 \), we have:
   \[
   4^{3/2} = 2^3 = 8
   \]

4. Therefore:
   \[
   4^{-3/2} = \frac{1}{8}
   \]

So, the answer is \( \frac{1}{8} \).

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you simplify other negative fractional exponents like \( 9^{-1/2} \)?
2. What are the general rules for working with fractional exponents?
3. How does \( 4^{3/2} \) differ from \( 4^{2/3} \), and how do you compute both?
4. How can negative exponents be applied in solving real-world problems?
5. What is the relationship between exponents and roots?

**Tip:** When simplifying fractional exponents, always try to break them into a root and a power for easier calculation.

Learn how to simplify the expression 4^(-3/2) step-by-step, covering key concepts such as negative exponents and fractional exponents. This problem illustrates the use of properties of exponents and radicals, making it suitable for students in Grades 9-10.

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