https://file.aiquickdraw.com/mathbot/file/1725421185880jskmjcvk.jpg

Let's work through the questions provided in the image you uploaded. 

### **12) Simplify and write as a positive exponent (when needed):**

a) \( 4x^{-2} \cdot 5x^{-3} \)  
To simplify, multiply the coefficients and add the exponents of \( x \):
\[
4 \times 5 = 20
\]
\[
x^{-2} \cdot x^{-3} = x^{-5}
\]
The result is \( 20x^{-5} \), and to write with a positive exponent:
\[
\text{Final Answer: } \frac{20}{x^5}
\]

b) \( a^{3x} \cdot a^{x^2} \)  
To simplify, add the exponents (since the bases are the same):
\[
a^{3x + x^2}
\]
No further simplification is possible.

c) \( \frac{x^2y^3}{y^2} \)  
Simplify by subtracting the exponents of \( y \):
\[
x^2 \cdot y^{3-2} = x^2 \cdot y
\]

d) \( 6x^2 + 8y \)  
This expression is already in its simplest form. 

e) \( 6y^2 \)  
This expression is already in its simplest form as well. 

### **13) Expand the brackets and simplify where possible:**

1. \( 4(x - 3) \)
\[
4x - 12
\]

2. \( 4(2x - 3) \)
\[
8x - 12
\]

3. \( 5x(2 - 7x) \)
\[
10x - 35x^2
\]

4. \( 3y(2 - y) \)
\[
6y - 3y^2
\]

5. \( 2(3p + 4) + 3(2p - 3) \)
Expand and combine like terms:
\[
6p + 8 + 6p - 9 = 12p - 1
\]

6. \( n(r + 3) + 2n \)
\[
nr + 3n + 2n = nr + 5n
\]

7. \( 2(2x + 5) - 4(x - 3) \)
Expand and simplify:
\[
4x + 10 - 4x + 12 = 22
\]

8. \( 2(x + 4) + 3(x - 3) \)
Expand and combine like terms:
\[
2x + 8 + 3x - 9 = 5x - 1
\]

9. \( 2(p + 2) + 3(2p - 3) \)
Expand and combine like terms:
\[
2p + 4 + 6p - 9 = 8p - 5
\]

10. \( 3(x - 5p) + 2(3p - 3) \)
Expand and combine like terms:
\[
3x - 15p + 6p - 6 = 3x - 9p - 6
\]

### **14) Solve for \( x \):**

a) \( 5x - 7 = 13 \)
\[
5x = 20 \quad \Rightarrow \quad x = 4
\]

b) \( 2x + 3 = 11 \)
\[
2x = 8 \quad \Rightarrow \quad x = 4
\]

c) \( 4x - 1 = 3x + 2 \)
\[
x = 3
\]

d) \( 2(3x + 4) = 16 \)
\[
6x + 8 = 16 \quad \Rightarrow \quad 6x = 8 \quad \Rightarrow \quad x = \frac{4}{3}
\]

e) \( \frac{3x}{4} + 2 = 5(2 + x) \)
Expand and simplify:
\[
3x + 8 = 40 + 20x \quad \Rightarrow \quad 17x = 32 \quad \Rightarrow \quad x = \frac{32}{17}
\]

f) \( 5(x - 1) = 7 \)
\[
5x - 5 = 7 \quad \Rightarrow \quad 5x = 12 \quad \Rightarrow \quad x = \frac{12}{5}
\]

Let me know if you want more detailed explanations or have any other questions! 

Here are five related questions:
1. What is the general rule for simplifying expressions with negative exponents?
2. How do you combine like terms after expanding an algebraic expression?
3. What is the process for solving linear equations with variables on both sides?
4. How can you verify your solutions to the equations?
5. What is the significance of the distributive property in expanding brackets?

**Tip:** Always check your solutions by substituting the value of \( x \) back into the original equation to ensure it satisfies the equation.

Simplify and write as a positive exponent (where needed) and expand the brackets and simplify where possible based on the provided image.

This page provides step-by-step solutions for simplifying exponents and expanding algebraic expressions. Concepts covered include exponentiation rules, distributive property, and combining like terms. Suitable for students in Grades 8-10.

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