Sure, let’s break down the steps for evaluating the given expression [tex]\( 3 \cdot (2 \cdot 8) + 7 \)[/tex] and identify the one that properly applies the commutative property of addition and the associative property of multiplication:
1. Starting Expression:
[tex]\[ 3 \cdot (2 \cdot 8) + 7 \][/tex]
2. Evaluate Inside Parentheses:
[tex]\[ 2 \cdot 8 = 16 \][/tex]
3. Substitute Back:
[tex]\[ 3 \cdot 16 + 7 \][/tex]
4. Perform Multiplication:
[tex]\[ 3 \cdot 16 = 48 \][/tex]
5. Add Constant:
[tex]\[ 48 + 7 = 55 \][/tex]
Thus, the original expression evaluates to 55.
Now, let's apply the commutative property of addition and the associative property of multiplication to rewrite the given expression:
1. Commutative Property of Addition allows us to rearrange terms:
[tex]\[ a + b = b + a \][/tex]
This property isn't immediately useful until we have an expression that involves addition besides the final constant, which we already have in the original form [tex]\(3 \cdot (2 \cdot 8) + 7\)[/tex].
2. Associative Property of Multiplication allows us to group factors differently:
[tex]\[ a \cdot (b \cdot c) = (a \cdot b) \cdot c \][/tex]
Applying this property, we get:
[tex]\[ 3 \cdot (2 \cdot 8) = (3 \cdot 2) \cdot 8 \][/tex]
Substitute back the evaluated expression:
[tex]\[ (3 \cdot 2) \cdot 8 + 7 = 6 \cdot 8 + 7 = 48 + 7 = 55 \][/tex]
Both expressions, [tex]\( 3 \cdot (2 \cdot 8) + 7 \)[/tex] and [tex]\( (3 \cdot 2) \cdot 8 + 7 \)[/tex], yield 55 when evaluated.
Thus, the expression that uses the commutative property of addition and the associative property of multiplication to rewrite the original expression [tex]\( 3 \cdot (2 \cdot 8) + 7 \)[/tex] is:
[tex]\[ (3 \cdot 2) \cdot 8 + 7 \][/tex]
