Absolutely! Let's go through each part step-by-step.
### Properties of Multiplication of Integers
1. Commutative Property: This property states that the order in which two numbers are multiplied does not change the product. Mathematically, if [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers, then:
[tex]\[
a \times b = b \times a
\][/tex]
2. Associative Property: This property states that the way in which numbers are grouped in multiplication does not change their product. If [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are integers, then:
[tex]\[
(a \times b) \times c = a \times (b \times c)
\][/tex]
3. Identity Property: This property states that any integer multiplied by 1 remains unchanged. If [tex]\( a \)[/tex] is an integer, then:
[tex]\[
a \times 1 = a
\][/tex]
4. Distributive Property: This property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately and then adding the results. If [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are integers, then:
[tex]\[
a \times (b + c) = a \times b + a \times c
\][/tex]
5. Zero Property: This property states that any integer multiplied by 0 is 0. If [tex]\( a \)[/tex] is an integer, then:
[tex]\[
a \times 0 = 0
\][/tex]
### Applying the BODMAS Rule to the Expression
Let's find the value of the expression [tex]\( -3 + \{(-4) \div 4 + 1\} + 3 \)[/tex].
According to the BODMAS/BIDMAS rule (Brackets, Orders (i.e., powers and roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right)):
1. Brackets: Resolve the expression inside the curly brackets [tex]\(\{\}\)[/tex].
2. BIDMAS inside the brackets:
- Division within the brackets:
[tex]\[
(-4) \div 4 = -1
\][/tex]
- Adding the result of the division to 1:
[tex]\[
-1 + 1 = 0
\][/tex]
Thus, the expression inside the curly brackets simplifies to 0.
3. Substituting back into the main expression:
[tex]\[
-3 + 0 + 3
\][/tex]
4. Adding and Subtracting (left-to-right):
- First, add [tex]\(-3\)[/tex] and 0:
[tex]\[
-3 + 0 = -3
\][/tex]
- Then, add [tex]\(-3\)[/tex] and 3:
[tex]\[
-3 + 3 = 0
\][/tex]
Therefore, the value of the given expression [tex]\( -3 + \{(-4) \div 4 + 1\} + 3 \)[/tex] is [tex]\( 0 \)[/tex].
