To determine which expression uses the commutative property to make it easier to evaluate the given expression [tex]\(\left(-\frac{1}{3}\right) \cdot \frac{2}{5} \cdot(-9)\)[/tex], let's analyze each of the provided options.
The commutative property of multiplication states that the order of factors can be changed without changing the product. So we can rearrange the factors in the original expression [tex]\(\left(-\frac{1}{3}\right) \cdot \frac{2}{5} \cdot (-9)\)[/tex] in different ways while the product remains the same.
Let's evaluate the given choices:
A. [tex]\(\left(-\frac{1}{3}\right) \cdot \frac{5}{2} \cdot (-9)\)[/tex]
This expression does not follow the commutative property correctly as the second factor has changed from [tex]\(\frac{2}{5}\)[/tex] to [tex]\(\frac{5}{2}\)[/tex].
B. [tex]\((-9) \cdot \frac{2}{5} \cdot \left(-\frac{1}{3}\right)\)[/tex]
Here, the order of the factors is changed, but each factor is the same as in the original expression. Indeed, this uses the commutative property properly.
C. [tex]\(\frac{1}{3} \cdot 5 \cdot \frac{2}{9}\)[/tex]
This expression not only changes the order of the factors but also modifies them. Specifically, [tex]\(-\frac{1}{3}\)[/tex] is turned into [tex]\(\frac{1}{3}\)[/tex], [tex]\(\frac{2}{5}\)[/tex] is eliminated and a 5 appears as a factor, and the [tex]\(-9\)[/tex] is re-expressed incorrectly as [tex]\(\frac{2}{9}\)[/tex]. This does not represent a correct application of the commutative property on the original expression.
D. [tex]\(\left(-\frac{1}{3}\right) \cdot (-9) \cdot \frac{2}{5}\)[/tex]
Here again, the order of the factors is changed, but each factor remains the same as in the original expression. This is a valid use of the commutative property.
Now, let's confirm which expressions correctly rearrange the terms without altering the factors.
Options B and D are valid since they maintain the integrity of each factor and only reorder them.
To be more precise, evaluating these two choices:
Original: [tex]\(\left(-\frac{1}{3}\right) \cdot \frac{2}{5} \cdot (-9)\)[/tex]
Reordered:
B. [tex]\((-9) \cdot \frac{2}{5} \cdot \left(-\frac{1}{3}\right)\)[/tex]
D. [tex]\(\left(-\frac{1}{3}\right) \cdot (-9) \cdot \frac{2}{5}\)[/tex]
Both rearrangements are valid and equivalent. Therefore, both expressions make it easier to evaluate using the commutative property.
So the final answer is:
B. [tex]\((-9) \cdot \frac{2}{5} \cdot \left(-\frac{1}{3}\right)\)[/tex]
and
D. [tex]\(\left(-\frac{1}{3}\right) \cdot (-9) \cdot \frac{2}{5}\)[/tex]
Thus, the correct choices are:
B and D.
