(2) $$ x^{\frac{5}{3}}=32 $$ (4) $$ x^{\frac{2}{5}}=4 $$ (6) $$ x^{\frac{5}{4}}=-32 $$ (8) $$ 40 x^{\frac{3}{4}}=5 $$ (10) $$ -16 x^{-\frac{2}{3}}+4=0 $$

Question

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Sure, let's solve each of the given equations step by step.

---

### **Problem (2)**

$$
x^{\frac{5}{3}} = 32
$$

**Solution:**

1. **Isolate $$ x $$:**
   
   To solve for $$ x $$, raise both sides of the equation to the reciprocal of $$ \frac{5}{3} $$, which is $$ \frac{3}{5} $$:

$$
   x = 32^{\frac{3}{5}}
   $$

2. **Calculate $$ 32^{\frac{3}{5}} $$:**
   
   - First, find the 5th root of 32. Since $$ 32 = 2^5 $$:
     
     $$
     32^{\frac{1}{5}} = 2
     $$
   
   - Then, raise this result to the 3rd power:
     
     $$
     (2)^3 = 8
     $$

**Answer:** $$ x = 8 $$

---

### **Problem (4)**

$$
x^{\frac{2}{5}} = 4
$$

**Solution:**

1. **Isolate $$ x $$:**
   
   Raise both sides to the reciprocal of $$ \frac{2}{5} $$, which is $$ \frac{5}{2} $$:

$$
   x = 4^{\frac{5}{2}}
   $$

2. **Calculate $$ 4^{\frac{5}{2}} $$:**
   
   - First, find the square root of 4:
     
     $$
     4^{\frac{1}{2}} = 2
     $$
   
   - Then, raise this result to the 5th power:
     
     $$
     (2)^5 = 32
     $$

**Answer:** $$ x = 32 $$

---

### **Problem (6)**

$$
x^{\frac{5}{4}} = -32
$$

**Solution:**

For real numbers, $$ x^{\frac{5}{4}} $$ implies taking the fourth root of $$ x $$ and then raising it to the 5th power.

- The fourth root of a real number is defined **only for non-negative** values of $$ x $$.
- Therefore, $$ x^{\frac{5}{4}} $$ is **non-negative** for all real $$ x $$.

Since the right side of the equation is negative ($$ -32 $$), there is **no real solution**.

**Answer:** No real solution exists.

---

### **Problem (8)**

$$
40 x^{\frac{3}{4}} = 5
$$

**Solution:**

1. **Isolate $$ x^{\frac{3}{4}} $$:**
   
   $$
   x^{\frac{3}{4}} = \frac{5}{40} = \frac{1}{8}
   $$

2. **Solve for $$ x $$:**
   
   Raise both sides to the reciprocal of $$ \frac{3}{4} $$, which is $$ \frac{4}{3} $$:

$$
   x = \left( \frac{1}{8} \right)^{\frac{4}{3}}
   $$

3. **Calculate $$ \left( \frac{1}{8} \right)^{\frac{4}{3}} $$:**
   
   - First, find the cube root of $$ \frac{1}{8} $$:
     
     $$
     \left( \frac{1}{8} \right)^{\frac{1}{3}} = \frac{1}{2}
     $$
   
   - Then, raise this result to the 4th power:
     
     $$
     \left( \frac{1}{2} \right)^4 = \frac{1}{16}
     $$

**Answer:** $$ x = \frac{1}{16} $$

---

### **Problem (10)**

$$
-16 x^{-\frac{2}{3}} + 4 = 0
$$

**Solution:**

1. **Isolate the term with $$ x $$:**
   
   $$
   -16 x^{-\frac{2}{3}} = -4
   $$
   
   Divide both sides by $$ -16 $$:

$$
   x^{-\frac{2}{3}} = \frac{1}{4}
   $$

2. **Rewrite $$ x^{-\frac{2}{3}} $$ as $$ \frac{1}{x^{\frac{2}{3}}} $$:**
   
   $$
   \frac{1}{x^{\frac{2}{3}}} = \frac{1}{4}
   $$
   
   Taking reciprocals on both sides:

$$
   x^{\frac{2}{3}} = 4
   $$

3. **Solve for $$ x $$:**
   
   Raise both sides to the reciprocal of $$ \frac{2}{3} $$, which is $$ \frac{3}{2} $$:

$$
   x = 4^{\frac{3}{2}}
   $$

4. **Calculate $$ 4^{\frac{3}{2}} $$:**
   
   - First, find the square root of 4:
     
     $$
     4^{\frac{1}{2}} = 2
     $$
   
   - Then, raise this result to the 3rd power:
     
     $$
     (2)^3 = 8
     $$

5. **Verification:**
   
   Substitute $$ x = 8 $$ back into the original equation:

$$
   -16 (8)^{-\frac{2}{3}} + 4 = -16 \left( \frac{1}{4} \right) + 4 = -4 + 4 = 0
   $$
   
   The equation holds true.

**Answer:** $$ x = 8 $$

---

### **Summary of Solutions**

- **(2) $$ x^{\frac{5}{3}} = 32 $$**  
  **Solution:** $$ x = 8 $$

- **(4) $$ x^{\frac{2}{5}} = 4 $$**  
  **Solution:** $$ x = 32 $$

- **(6) $$ x^{\frac{5}{4}} = -32 $$**  
  **Solution:** No real solution exists.

- **(8) $$ 40 x^{\frac{3}{4}} = 5 $$**  
  **Solution:** $$ x = \frac{1}{16} $$

- **(10) $$ -16 x^{-\frac{2}{3}} + 4 = 0 $$**  
  **Solution:** $$ x = 8 $$

---

If you have any further questions or need more explanations on any of the steps, feel free to ask!
Here are the solutions to the equations: 1. $$ x^{\frac{5}{3}} = 32 $$ **Answer:** $$ x = 8 $$ 2. $$ x^{\frac{2}{5}} = 4 $$ **Answer:** $$ x = 32 $$ 3. $$ x^{\frac{5}{4}} = -32 $$ **Answer:** No real solution 4. $$ 40 x^{\frac{3}{4}} = 5 $$ **Answer:** $$ x = \frac{1}{16} $$ 5. $$ -16 x^{-\frac{2}{3}} + 4 = 0 $$ **Answer:** $$ x = 8 $$

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