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The image shows a set of mathematical expressions that need to be converted into their equivalent rational exponent form. Let's go through each one:

1. **\( \sqrt[5]{2^2} \)**

   The general form of a radical expression is \( \sqrt[n]{a^m} \), which can be written in rational exponent form as \( a^{\frac{m}{n}} \).

   So, \( \sqrt[5]{2^2} \) can be written as \( 2^{\frac{2}{5}} \).

2. **\( \frac{1}{\sqrt[6]{3^5}} \)**

   This expression can be written as \( 3^{-\frac{5}{6}} \).

3. **\( \sqrt[3]{5^{-2}} \)**

   Using the same rule, \( \sqrt[3]{5^{-2}} \) can be written as \( 5^{-\frac{2}{3}} \).

4. **\( \frac{1}{\sqrt[5]{4^2}} \)**

   This can be written as \( 4^{-\frac{2}{5}} \).

5. **\( \sqrt[4]{\frac{1}{7^3}} \)**

   First, simplify the expression inside the radical:
   \[
   \sqrt[4]{\frac{1}{7^3}} = \sqrt[4]{7^{-3}}
   \]
   Then, it can be written as \( 7^{-\frac{3}{4}} \).

Would you like more detailed explanations or have any further questions? Here are some related questions to consider:

1. What are the rules for converting any general radical expression into a rational exponent?
2. How do the negative exponents affect the placement of terms in a fraction?
3. Can you simplify an expression after converting it to a rational exponent form?
4. How do you handle expressions with both positive and negative rational exponents?
5. What are the benefits of using rational exponents over radical notation in more advanced mathematics?

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

Convert the following radical expressions into rational exponent form: 1) 5th root of (2^2), 2) 1 over 6th root of (3^5), 3) 3rd root of (5^-2), 4) 1 over 5th root of (4^2), 5) 4th root of (1 over 7^3)

This guide shows how to convert complex radical expressions like 5th root of (2^2), 1 over 6th root of (3^5), and others into rational exponent form, covering key concepts in algebra and exponent rules. Suitable for students in Grades 9-12.

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