Math worksheet focusing on algebraic expressions and multiplication properties.

Let's solve each part of Question 3 and Question 4 step by step.

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✔ Question 3

Given:
$a = 3$, $b = -4$, $c = -2$

#### a) $a + b$
$$
a + b = 3 + (-4) = 3 - 4 = -1
$$

#### b) $b + c$
$$
b + c = (-4) + (-2) = -4 - 2 = -6
$$

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#### Find the product for each:

##### [a] $-3 \times 5$
$$
-3 \times 5 = -15
$$

##### [b] $(-4) \times (-6)$
Negative × Negative = Positive:
$$
(-4) \times (-6) = 24
$$

##### [c] $-9 \times 0$
Any number times zero is zero:
$$
-9 \times 0 = 0
$$

##### [d] $6 \times (-3)$
Positive × Negative = Negative:
$$
6 \times (-3) = -18
$$

##### [e] $(-100) \times |-2|$
First, find absolute value: $|-2| = 2$
Then:
$$
(-100) \times 2 = -200
$$

##### [f] $(-31) \times 3$
$$
(-31) \times 3 = -93
$$

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✔ Question 4

#### Write the property of multiplication in the set $\mathbb{Z}$ (integers):

##### a) $-12 \times 1 = -12$
This shows that multiplying any number by 1 gives the same number.
➡️ Multiplicative Identity Property

##### b) $-5 \times (9 \times 7) = (-5 \times 9) \times 7$
This shows grouping doesn't matter in multiplication.
➡️ Associative Property of Multiplication

##### c) $5 \times (-2) = (-2) \times 5$
Order doesn’t matter — commutative.
➡️ Commutative Property of Multiplication

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#### Complete:

##### [a] $3^2 = ?$
$$
3^2 = 3 \times 3 = 9
$$

##### [b] $(-7)^2 = ?$
$$
(-7)^2 = (-7) \times (-7) = 49
$$

##### [c] $2^3 \times 2^3 = ?$
Use exponent rule: $a^m \times a^n = a^{m+n}$
$$
2^3 \times 2^3 = 2^{3+3} = 2^6 = 64
$$

Alternatively: $8 \times 8 = 64$

##### [d] $(-3)^2 \times (-3)^3 = ?$
Again, use exponent rules:
$$
(-3)^2 \times (-3)^3 = (-3)^{2+3} = (-3)^5
$$
Now compute:
$$
(-3)^5 = -243
$$
(Since odd power of negative number is negative)

##### [e] $2^2 \times (-2)^2 = ?$
First: $2^2 = 4$, $(-2)^2 = 4$
So:
$$
4 \times 4 = 16
$$

Note: Even though bases are different signs, squaring makes both positive.

##### [f] $3^2 \times (-3)^3 = ?$
Compute each:
- $3^2 = 9$
- $(-3)^3 = -27$

Now multiply:
$$
9 \times (-27) = -243
$$

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✔ Final Answers:

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Question 3

- a) $a + b = -1$
- b) $b + c = -6$

Products:
- [a] $-3 \times 5 = -15$
- [b] $(-4) \times (-6) = 24$
- [c] $-9 \times 0 = 0$
- [d] $6 \times (-3) = -18$
- [e] $(-100) \times |-2| = -200$
- [f] $(-31) \times 3 = -93$

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Question 4

Properties:
- a) Multiplicative Identity
- b) Associative Property
- c) Commutative Property

Complete:
- [a] $3^2 = 9$
- [b] $(-7)^2 = 49$
- [c] $2^3 \times 2^3 = 64$
- [d] $(-3)^2 \times (-3)^3 = -243$
- [e] $2^2 \times (-2)^2 = 16$
- [f] $3^2 \times (-3)^3 = -243$

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✔ All problems solved! Let me know if you'd like this formatted as a worksheet answer sheet.