Question

write an equivalent expression
19\left(d^{2} + 1.03\
ight)
select all that apply.
a. 1.03\left(d^{2} + 19\
ight)
b. 19d^{2} + 1.03
c. \left(d^{2} + 1.03\
ight)19

Explanation:

Step1: Recall the commutative property of multiplication

The commutative property of multiplication states that \(a\times b = b\times a\). So, for the expression \(19(d^{2}+1.03)\), we can rewrite it as \((d^{2}+1.03)\times19\) or \((d^{2}+1.03)19\) (since multiplication is implied when a number is next to a parenthesis).

Step2: Analyze Option A

For option A: \(1.03(d^{2}+19)\). Using the distributive property, this would be \(1.03d^{2}+1.03\times19\), which is not equivalent to \(19d^{2}+19\times1.03\) (the expansion of the original expression). So A is incorrect.

Step3: Analyze Option B

For option B: \(19d^{2}+1.03\). The original expression, when expanded using the distributive property \(a(b + c)=ab+ac\), should be \(19\times d^{2}+19\times1.03=19d^{2}+19.57\), not \(19d^{2}+1.03\). So B is incorrect.

Step4: Analyze Option C

For option C: \((d^{2}+1.03)19\). By the commutative property of multiplication, \(19(d^{2}+1.03)=(d^{2}+1.03)\times19=(d^{2}+1.03)19\). So C is equivalent.

Answer:

C. \((d^{2}+1.03)19\)