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Sagot :
To determine which option shows the best use of the associative and commutative properties for simplifying the expression [tex]\(\frac{2}{5} - 15 + 8 - \frac{1}{5} + 3\)[/tex], let's carefully examine each option.
### Option A
[tex]\[ \left[\frac{2}{5} + 8 + (-15)\right] + \left[3 + \left(-\frac{1}{5}\right)\right] \][/tex]
### Option B
[tex]\[ \left[3 + \frac{2}{5} + (-15)\right] + \left[\left(-\frac{1}{5}\right) + 8\right] \][/tex]
### Option C
[tex]\[ \left[\frac{2}{5} + (-15)\right] + \left[8 + \left(-\frac{1}{5}\right) + 3\right] \][/tex]
### Option D
[tex]\[ \left[\frac{2}{5} + \left(-\frac{1}{5}\right)\right] + \left[8 + 3 + (-15)\right] \][/tex]
To identify the best use of associative and commutative properties, we can check how the grouping of terms makes simplification easier:
#### Option A
Grouping [tex]\(\frac{2}{5} + 8 + (-15)\)[/tex] and [tex]\(3 + (-\frac{1}{5})\)[/tex] does help somewhat, but it does not primarily group the similar fraction terms together for easier cancellation.
#### Option B
Grouping [tex]\(3 + \frac{2}{5} + (-15)\)[/tex] and [tex]\(\left(-\frac{1}{5}\right) + 8\)[/tex] again does not prioritize simplifying similar fractional terms first.
#### Option C
Grouping [tex]\(\frac{2}{5} + (-15)\)[/tex] and [tex]\(\left[8 + \left(-\frac{1}{5}\right) + 3\right]\)[/tex] does not make simplifying the fractions straightforward.
#### Option D
Grouping [tex]\(\frac{2}{5} + \left(-\frac{1}{5}\right)\)[/tex] and [tex]\(\left[8 + 3 + (-15)\right]\)[/tex]:
1. [tex]\(\frac{2}{5} + \left(-\frac{1}{5}\right) = \frac{2}{5} - \frac{1}{5} = \frac{1}{5}\)[/tex]
2. [tex]\(8 + 3 + (-15) = 8 + 3 - 15 = -4\)[/tex]
So, by simplifying:
[tex]\[ \frac{1}{5} + (-4) = -3.8 \][/tex]
Therefore, the grouping in option D simplifies the expression most effectively by using the associative and commutative properties to simplify the fractions first and then the integer terms. Therefore, the best answer is:
[tex]\[ \boxed{D} \][/tex]
### Option A
[tex]\[ \left[\frac{2}{5} + 8 + (-15)\right] + \left[3 + \left(-\frac{1}{5}\right)\right] \][/tex]
### Option B
[tex]\[ \left[3 + \frac{2}{5} + (-15)\right] + \left[\left(-\frac{1}{5}\right) + 8\right] \][/tex]
### Option C
[tex]\[ \left[\frac{2}{5} + (-15)\right] + \left[8 + \left(-\frac{1}{5}\right) + 3\right] \][/tex]
### Option D
[tex]\[ \left[\frac{2}{5} + \left(-\frac{1}{5}\right)\right] + \left[8 + 3 + (-15)\right] \][/tex]
To identify the best use of associative and commutative properties, we can check how the grouping of terms makes simplification easier:
#### Option A
Grouping [tex]\(\frac{2}{5} + 8 + (-15)\)[/tex] and [tex]\(3 + (-\frac{1}{5})\)[/tex] does help somewhat, but it does not primarily group the similar fraction terms together for easier cancellation.
#### Option B
Grouping [tex]\(3 + \frac{2}{5} + (-15)\)[/tex] and [tex]\(\left(-\frac{1}{5}\right) + 8\)[/tex] again does not prioritize simplifying similar fractional terms first.
#### Option C
Grouping [tex]\(\frac{2}{5} + (-15)\)[/tex] and [tex]\(\left[8 + \left(-\frac{1}{5}\right) + 3\right]\)[/tex] does not make simplifying the fractions straightforward.
#### Option D
Grouping [tex]\(\frac{2}{5} + \left(-\frac{1}{5}\right)\)[/tex] and [tex]\(\left[8 + 3 + (-15)\right]\)[/tex]:
1. [tex]\(\frac{2}{5} + \left(-\frac{1}{5}\right) = \frac{2}{5} - \frac{1}{5} = \frac{1}{5}\)[/tex]
2. [tex]\(8 + 3 + (-15) = 8 + 3 - 15 = -4\)[/tex]
So, by simplifying:
[tex]\[ \frac{1}{5} + (-4) = -3.8 \][/tex]
Therefore, the grouping in option D simplifies the expression most effectively by using the associative and commutative properties to simplify the fractions first and then the integer terms. Therefore, the best answer is:
[tex]\[ \boxed{D} \][/tex]
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