https://file.aiquickdraw.com/mathbot/file/1725896240254zqlkrzpp.jpg

In this image, you are asked to identify which expressions are mathematically meaningful or defined.

Let's analyze each expression one by one:

1. **\((-64)^{\frac{2}{5}}\)**:
   - Raising a negative number to a fractional power where the denominator is odd is defined. Specifically, \( (-64)^{\frac{2}{5}} \) can be calculated because 64 has a fifth root, and the exponent \( \frac{2}{5} \) results in a real number. So, this expression **has meaning**.

2. **\(\left(\frac{1}{2}\right)^{\frac{2}{5}}\)**:
   - Raising a positive fraction to any real power is always defined, so \(\left(\frac{1}{2}\right)^{\frac{2}{5}}\) is **meaningful**.

3. **\((-25)^{-\frac{1}{2}}\)**:
   - Raising a negative number to a fractional exponent with an even denominator (like \(-\frac{1}{2}\)) results in a complex number, not a real one. Therefore, \((-25)^{-\frac{1}{2}}\) is **undefined in the real number system**.

4. **\(\left(\frac{5}{8}\right)^{-\frac{8}{5}}\)**:
   - Raising a positive fraction to any real exponent is always defined. So, \(\left(\frac{5}{8}\right)^{-\frac{8}{5}}\) is **meaningful**.

5. **\(5^{-\frac{8}{5}}\)**:
   - Raising a positive integer to any real exponent is also always defined, so \(5^{-\frac{8}{5}}\) is **meaningful**.

### Final Answer:
The following expressions **have meaning**:
- \((-64)^{\frac{2}{5}}\)
- \(\left(\frac{1}{2}\right)^{\frac{2}{5}}\)
- \(\left(\frac{5}{8}\right)^{-\frac{8}{5}}\)
- \(5^{-\frac{8}{5}}\)

The expression \((-25)^{-\frac{1}{2}}\) **does not have meaning** in the real number system.

Would you like further clarification or detailed calculations?

Here are 5 related questions to explore further:
1. What is the difference between real and complex numbers when raising negative bases to fractional powers?
2. How do you simplify expressions with fractional exponents?
3. What is the fifth root of a number and how do you calculate it?
4. How does raising a number to a negative exponent affect its value?
5. Why are even roots of negative numbers not defined in the real number system?

**Tip**: When working with fractional exponents, remember that the denominator represents the root and the numerator represents the power.

Choose expressions that are meaningful from the following list of fractional exponent expressions involving negative and positive bases.

Analyze expressions with fractional exponents and determine which ones are defined. This problem involves key concepts like fractional powers, roots of numbers, and the difference between real and complex number operations, suitable for Grades 9-12.

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