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Let's solve the exercises step by step:

### 1. For each of the following, insert =, >, or <:
- (a) \(-3 \_\_ 3\)  
  \(-3 < 3\)
  
- (b) \(-10 \div 2 \_\_ 5\)  
  \(-10 \div 2 = -5\) so \(-5 < 5\)
  
- (c) \(20 \times (-1) \_\_ \frac{-40}{-2}\)  
  \(20 \times (-1) = -20\) and \(\frac{-40}{-2} = 20\), so \(-20 < 20\)

### 2. Calculate:
- (a) \(-5 + (-8)\)  
  \(-5 + (-8) = -13\)
  
- (b) \(12 - 96\)  
  \(12 - 96 = -84\)
  
- (c) \(-12 - 96\)  
  \(-12 - 96 = -108\)
  
- (d) \(-4 - 8 - 9\)  
  \(-4 - 8 - 9 = -21\)
  
- (e) \(-12 + 96\)  
  \(-12 + 96 = 84\)
  
- (f) \(-7 - (-7)\)  
  \(-7 - (-7) = 0\)

### 3. Find:
- (a) \(-6 \times 4\)  
  \(-6 \times 4 = -24\)
  
- (b) \(-9 \times 8 \times (-1)\)  
  \(-9 \times 8 = -72\), and \(-72 \times (-1) = 72\)
  
- (c) \((-12)^2\)  
  \((-12)^2 = 144\)
  
- (d) \(\frac{-9 \times (-7)}{3}\)  
  \(-9 \times (-7) = 63\), and \(\frac{63}{3} = 21\)
  
- (e) \(-150 \div (-2 - 3)\)  
  \(-150 \div (-5) = 30\)
  
- (f) \(-10 + 7 \times (-3)\)  
  \(7 \times (-3) = -21\), and \(-10 + (-21) = -31\)

### 4. State whether the answer is positive or negative:
- (a) \(-3 \times (-3) \times (-3)\)  
  Answer: Negative (since the product of three negative numbers is negative)

- (b) \(-109 \times 142 \times (-83)\)  
  Answer: Positive (since the product of two negatives is positive, and then multiplying by a positive number)

- (c) \(-2 \times (-1 - (-3))\)  
  \(-1 - (-3) = -1 + 3 = 2\), so \(-2 \times 2 = -4\)  
  Answer: Negative

### 5. Copy and complete:
- (a) \(\_\_ + 9 = -6\)  
  Answer: \(-15\), since \(-15 + 9 = -6\)
  
- (b) \(\_\_ \times (-3) = -6 \times (-4)\)  
  \(-6 \times (-4) = 24\), so \(\_\_ = \frac{24}{-3} = -8\)
  
- (c) \(16 \times \_\_ = -64\)  
  Answer: \(\_\_ = -4\), since \(16 \times (-4) = -64\)

### 6. If \(a = 6\) and \(b = -4\), find the value of:
- (a) \(-a + b\)  
  \(-a + b = -6 + (-4) = -10\)
  
- (b) \(a - b\)  
  \(a - b = 6 - (-4) = 6 + 4 = 10\)
  
- (c) \(2(b - a)\)  
  \(b - a = -4 - 6 = -10\), so \(2 \times (-10) = -20\)
  
- (d) \(-ab^2\)  
  \(b^2 = (-4)^2 = 16\), so \(-ab^2 = -(6 \times 16) = -96\)
  
- (e) \(a^2 + b^2\)  
  \(a^2 = 6^2 = 36\) and \(b^2 = 16\), so \(a^2 + b^2 = 36 + 16 = 52\)
  
- (f) \(24 \div (ab)\)  
  \(ab = 6 \times (-4) = -24\), so \(24 \div (-24) = -1\)

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Would you like further clarification or details on any of these solutions? Here are 5 related questions to expand on the topics:

1. What is the rule for multiplying and dividing negative numbers?
2. How does the order of operations affect expressions with negative numbers?
3. Why does raising a negative number to an even power result in a positive number?
4. Can you explain how to simplify expressions with multiple negative signs?
5. What is the effect of negative exponents on powers?

**Tip**: When multiplying or dividing negative numbers, an odd number of negative factors results in a negative product, while an even number of negative factors results in a positive product.

This exercise set involves solving arithmetic and algebraic problems with negative numbers. Topics include basic addition, subtraction, multiplication, and division of negative numbers, as well as solving simple algebraic equations. Suitable for Grades 6-8.

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