https://file.aiquickdraw.com/mathbot/file/1726089869853eved5ljo.jpg

I can help review the solutions from the image you uploaded.

### Let's go through the problems:

#### **Top Section: Evaluate each expression for the given value**

1. \( 5x^{\frac{1}{4}} \) for \( x = 16 \)
   - Correct process:
     \[
     5(16^{\frac{1}{4}}) = 5(2) = 10
     \]
   - However, in the image, the value of \( \sqrt[4]{16} \) is incorrectly noted. The correct answer is 10, not 80.

2. \( x^{-\frac{2}{3}} \) for \( x = 64 \)
   - Correct process:
     \[
     64^{-\frac{2}{3}} = \left(\sqrt[3]{64}\right)^{-2} = 4^{-2} = \frac{1}{16}
     \]
   - The solution is correct.

3. \( \frac{x}{\sqrt[7]{-x}} \) for \( x = -1 \)
   - Correct process:
     \[
     \frac{-1}{\sqrt[7]{1}} = \frac{-1}{1} = -1
     \]
   - The answer should be \( -1 \), but the work shows \( -1/\sqrt[7]{-x} \), which may be correct as long as the final evaluation leads to \( -1 \).

#### **Middle Section: Change each exponential expression to radical form**

4. \( x^{\frac{2}{3}} \)
   - Correct conversion:
     \[
     \sqrt[3]{x^2}
     \]
   - This seems fine based on the image.

5. \( 4x^{\frac{2}{7}} \)
   - Correct conversion:
     \[
     4\sqrt[7]{x^2}
     \]
   - The answer looks correct.

6. \( 3(x + 2)^{\frac{2}{5}} \)
   - Correct conversion:
     \[
     3\sqrt[5]{(x+2)^2}
     \]
   - The image shows the correct form.

#### **Bottom Section: Change each radical expression to exponential form**

7. \( \sqrt[3]{x} \)
   - Correct conversion:
     \[
     x^{\frac{1}{3}}
     \]
   - This is correct as shown in the image.

8. \( \frac{1}{\sqrt{x+1}} \)
   - Correct conversion:
     \[
     (x+1)^{-\frac{1}{2}}
     \]
   - This is correct.

9. \( \frac{x-2}{\sqrt{x+1}} \)
   - Correct conversion:
     \[
     (x-2)(x+1)^{-\frac{1}{2}}
     \]
   - The final answer is correct based on the image.

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### Would you like me to go over any of these in more detail? 

### 5 Related Questions:
1. What is the general method for converting fractional exponents to radicals?
2. How can negative exponents be simplified further in expressions?
3. What happens if the base of a power is negative, as in problem 3?
4. Why does \( \sqrt[3]{x} \) have the exponent \( \frac{1}{3} \)?
5. How can radical and exponential forms help in solving equations?

### Tip:
Always check whether the exponent or root applies to the entire term or just the variable, as this affects the final simplification.

Evaluate each expression for the given value and change exponential expressions to radical form, as shown in the uploaded image.

This exercise involves evaluating exponential expressions for given values, converting exponential expressions into radical form, and vice versa. It covers the key concepts of exponents, radicals, and algebraic manipulation for high school students in Grades 9-12.

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