Let's examine each option to determine which correctly demonstrates the use of the Commutative Property of Multiplication. The Commutative Property of Multiplication states that [tex]\(a \times b = b \times a\)[/tex]; in other words, changing the order of the factors does not change the product.
### Option 1:
[tex]\[ 3 \left(b^{10} + 4\right) = 3 \left(4 + b^{10}\right) \][/tex]
This is correct because multiplication by 3 on both sides of [tex]\(b^{10} + 4\)[/tex] results in the same expression no matter the ordering within the parentheses:
[tex]\[ 3 \left(b^{10} + 4\right) = 3 \left(4 + b^{10}\right) \][/tex]
This expression confirms the property since it doesn't matter if you add the terms inside the parentheses in a different order.
### Option 2:
[tex]\[ 11 \left(b + z\right) = 11 b + 11 z \][/tex]
This option shows the Distributive Property, not the Commutative Property of Multiplication. The correct demonstration of the Commutative Property would involve switching the places of the multiplicands only. This expression reveals how multiplication distributes over addition.
### Option 3:
[tex]\[ z^{11} \cdot 2 = z^{11} + 2 \][/tex]
This is incorrect because it incorrectly represents multiplication and addition equivalence, which is not allowed by the Commutative Property. The Commutative Property relates to the switching of terms around multiplication, not their transformation into addition.
### Option 4:
[tex]\[ 2 \left(b^{10} + z^{11}\right) = \left(b^{10} + z^{11}\right) \cdot 2 \][/tex]
This is correct because it demonstrates that multiplying the sum [tex]\(b^{10} + z^{11}\)[/tex] by 2 results in the same product whether 2 is multiplied before or after the sum:
[tex]\[ 2 \left(b^{10} + z^{11}\right) = \left(b^{10} + z^{11}\right) \cdot 2 \][/tex]
This is an application of the Commutative Property of Multiplication since reversing the order of the multiplication operation does not change the result.
After examining all the options, it is evident that Option 4 is the correct demonstration of the Commutative Property of Multiplication.
