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 $24 \div (-24) = -1$

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.