https://file.aiquickdraw.com/mathbot/file/1726533862072gadt4ddx.jpg

The problem in the image requires selecting equivalent expressions for given exponential forms. Let's go through the expressions one by one and match them with the possible options.

### Expressions to Evaluate:
1. \( \left(\frac{2}{3}\right)^4 \)
2. \( 1 \)
3. \( \frac{2}{3} \)
4. \( (-0.4)^4 \)

### Matching with Options:

#### First Column \( \left(\frac{2}{3}\right)^4 \):
- This is a simple exponential expression, where both 2 and 3 are raised to the 4th power. We get:
  \[
  \left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} = \frac{16}{81}
  \]
- We need to find a match in the table for \( \frac{16}{81} \).

#### Second Column \( 1 \):
- \( 1^x = 1 \) for any value of \( x \), so the second column corresponds to any expression that simplifies to 1.

#### Third Column \( \frac{2}{3} \):
- This is just the fraction \( \frac{2}{3} \), no exponentiation involved. We match this with a simplified form in the table.

#### Fourth Column \( (-0.4)^4 \):
- Calculating \( (-0.4)^4 \), since the exponent is even, the result is positive:
  \[
  (-0.4)^4 = 0.4^4 = (0.4 \times 0.4 \times 0.4 \times 0.4) = 0.0256
  \]
- We find the corresponding value in the table.

### Now, we will fill the table based on these evaluations.

Let me know if you'd like further clarification or any detailed calculations!

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**Relative Questions:**
1. What is the general rule for simplifying exponents of fractions?
2. How does raising negative numbers to even or odd exponents affect the result?
3. Why does \( 1^x \) always equal 1 regardless of the exponent?
4. How do fractional exponents differ from integer exponents?
5. What happens to the base when negative numbers are raised to different powers?

**Tip:** When dealing with fractions raised to powers, both the numerator and the denominator are raised to the power separately.

Select the equivalent expression for each exponential form given in the table: (2/3)^4, 1, 2/3, (-0.4)^4.

Solve exponential expression problems such as (2/3)^4, 1, and (-0.4)^4 by identifying equivalent forms. Learn about the properties of exponents and how to simplify expressions involving fractions and negative numbers.

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