https://file.aiquickdraw.com/mathbot/file/1734365347499bqxx29z2.jpg

Let’s break down the math problems in the image step-by-step.

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### 1. Problems Overview:
The image contains 9 distinct math problems. They cover:
- **Exponents and operations with powers**
- **Simplifications**
- **Solving for \( x \)**
- **Working with fractions and exponents**

Below is a **solution outline** for each problem:

---

### **Problem 1**
Simplify:
\[
\left( \frac{3}{2} \right)^2 + \left( \frac{5}{3} \right)^2 + \left( \frac{4}{3} \right)^2
\]
- Square each term and sum the results.

---

### **Problem 2**
Simplify the product:
\[
(3^x)^2 \cdot (3^x)^3 \cdot (3^x)^4
\]
- Use the exponent multiplication rule: \( a^m \cdot a^n = a^{m+n} \).

---

### **Problem 3**
Simplify the expression:
\[
\left( \frac{3}{2} \right)^1 \cdot \left( \frac{1}{2} \right)^1 \cdot x^1 = 6x^2
\]
- Solve for \( x \).

---

### **Problem 4 (a)**
Solve for \( x \) in:
\[
5^x + 5^{-x} = 5^2
\]
- Use substitutions to simplify.

---

### **Problem 4 (b)**
Solve for \( x \) in:
\[
\frac{2x - 2}{3x - 5} = \frac{1}{9}
\]
- Cross-multiply and solve for \( x \).

---

### **Problem 4 (c)**
Simplify:
\[
\left( \frac{3}{2} \right)^{x+1} - \left( \frac{3}{2} \right)^x = \left( \frac{3}{2} \right)^1
\]
- Factor out common powers.

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### **Problem 4 (d)**
Simplify and solve for \( x \):
\[
2 \cdot \left( \frac{1}{2} \right)^{x+1} = \left( \frac{1}{2} \right)^x
\]
- Apply exponent rules.

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### **Problem 5**
Determine the number \( k \) such that:
\[
\left( \frac{3}{4} \right)^k \cdot N = \left( \frac{3}{4} \right)^7
\]
- Solve for \( N \).

---

### **Problem 6**
Simplify:
\[
\frac{2 \cdot 10^{-2} + 1.25}{5 \cdot 10^{-2}}
\]
- Combine terms in the numerator and divide.

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Would you like me to **solve any specific problem in detail** or all of them step-by-step? Let me know which part you want me to focus on!

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### **5 Related Questions**
1. How do you simplify expressions with powers and roots?
2. How do you solve exponential equations like \( a^x + a^{-x} = k \)?
3. What are the rules for solving rational equations?
4. How do you factor terms with common powers?
5. How do you simplify complex fractions with exponents?

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**Tip**: When dealing with powers, always group terms with the same base and apply exponent rules carefully!

Solve a series of math problems involving exponents, fractions, and equations as shown in the provided image.

Explore step-by-step solutions to problems involving exponents, rational equations, and algebraic simplifications. This guide covers exponent rules, solving linear equations, and factoring techniques, perfect for Grades 9-11 students.

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