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Let’s work through each of these problems step-by-step.

---

1. **Solve \( (x + 9)^4 = 10,000 \)**

   Take the fourth root of both sides:
   \[
   x + 9 = \sqrt[4]{10,000}
   \]
   Calculate \( \sqrt[4]{10,000} \):
   \[
   \sqrt[4]{10,000} = 10
   \]
   So,
   \[
   x + 9 = 10 \Rightarrow x = 10 - 9 = 1
   \]

   **Answer**: \( x = 1 \)

---

2. **Solve \( x^6 = -56 \)**

   Since any real number raised to an even power results in a non-negative value, \( x^6 = -56 \) has no real solutions.

   **Answer**: No real solutions.

---

3. **Evaluate \( 100 \cdot \sqrt[3]{0.001} - \sqrt[5]{-32} - (-4\sqrt{2})^2 \)**

   - First, find \( \sqrt[3]{0.001} \):
     \[
     \sqrt[3]{0.001} = 0.1
     \]
     So,
     \[
     100 \cdot 0.1 = 10
     \]

   - Next, find \( \sqrt[5]{-32} \):
     \[
     \sqrt[5]{-32} = -2
     \]

   - Now, calculate \( (-4\sqrt{2})^2 \):
     \[
     (-4\sqrt{2})^2 = 16 \cdot 2 = 32
     \]

   Substitute these results:
   \[
   10 - (-2) - 32 = 10 + 2 - 32 = -20
   \]

   **Answer**: \( -20 \)

---

4. **Evaluate \( \sqrt[3]{54} - 3 \cdot \sqrt[3]{16} + 5 \cdot \sqrt[3]{128} + \sqrt[3]{2000} \)**

   - First, calculate each cube root individually:
     \[
     \sqrt[3]{54} \approx 3.78
     \]
     \[
     \sqrt[3]{16} \approx 2.52
     \]
     \[
     \sqrt[3]{128} = 5
     \]
     \[
     \sqrt[3]{2000} \approx 12.6
     \]

   Substitute these values into the expression:
   \[
   3.78 - 3 \cdot 2.52 + 5 \cdot 5 + 12.6
   \]
   Calculate each term:
   \[
   3.78 - 7.56 + 25 + 12.6 = 33.82
   \]

   **Answer**: \( 33.82 \) (approximately)

---

5. **Solve \( \sqrt{3x + 1} - \sqrt{x + 1} = 2 \)**

   Let \( y = \sqrt{x + 1} \), so \( y^2 = x + 1 \) and \( x = y^2 - 1 \).

   Substitute this into the equation:
   \[
   \sqrt{3(y^2 - 1) + 1} - y = 2
   \]
   Simplify and solve for \( y \) to find \( x \).

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Would you like further details on any of these steps, or have any specific questions? 

### Related Questions

1. How can we handle equations with roots of negative numbers in different contexts?
2. What methods can we use to simplify equations involving nested roots?
3. How can the concept of imaginary numbers help in solving \( x^6 = -56 \)?
4. What are the limitations of real numbers in equations with roots?
5. How does substitution work in simplifying complex expressions?

**Tip**: Remember to check for extraneous solutions when squaring both sides of an equation.

1) (x + 9)^4 = 10,000 
2) x^6 = -56 
3) 100 * cube root(0.001) - fifth root(-32) - (-4 * sqrt(2))^2 
4) cube root(54) - 3 * cube root(16) + 5 * cube root(128) + cube root(2000) 
5) sqrt(3x + 1) - sqrt(x + 1) = 2

Explore detailed solutions to advanced algebraic equations involving exponents, roots, and radicals. Learn how to handle real and complex roots, simplify expressions, and apply algebraic properties to solve equations, such as (x + 9)^4 = 10,000 and x^6 = -56. Suitable for high school students.

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