Answer :
Let's verify each of the given properties for the values [tex]\(a = \frac{-7}{3}\)[/tex], [tex]\(b = \frac{5}{-4}\)[/tex], and [tex]\(c = \frac{1}{2}\)[/tex].
### Property (a): [tex]\(a + (b + c) = (a + b) + c\)[/tex]
We need to check if the associative property of addition holds.
1. Calculate [tex]\(b + c\)[/tex]:
[tex]\[ b + c = \frac{5}{-4} + \frac{1}{2} = \frac{-5}{4} + \frac{1}{2} = \frac{-5}{4} + \frac{2}{4} = \frac{-3}{4} \][/tex]
2. Calculate [tex]\(a + (b + c)\)[/tex]:
[tex]\[ a + (b + c) = \frac{-7}{3} + \frac{-3}{4} = \frac{-28}{12} + \frac{-9}{12} = \frac{-37}{12} \approx -3.0833333333333335 \][/tex]
3. Calculate [tex]\(a + b\)[/tex]:
[tex]\[ a + b = \frac{-7}{3} + \frac{5}{-4} = \frac{-28}{12} + \frac{-15}{12} = \frac{-43}{12} \][/tex]
4. Calculate [tex]\((a + b) + c\)[/tex]:
[tex]\[ (a + b) + c = \frac{-43}{12} + \frac{1}{2} = \frac{-43}{12} + \frac{6}{12} = \frac{-37}{12} \approx -3.0833333333333335 \][/tex]
Therefore:
[tex]\[ a + (b + c) = \frac{-37}{12} \quad \text{and} \quad (a + b) + c = \frac{-37}{12} \][/tex]
Since both sides are equal:
[tex]\[ a + (b + c) = (a + b) + c \][/tex]
### Property (b): [tex]\(a \times (b \times c) = a \times (b \times c)\)[/tex]
This property seems straightforward as it involves the same exact terms on both sides.
Calculate [tex]\(b \times c\)[/tex]:
[tex]\[ b \times c = \frac{5}{-4} \times \frac{1}{2} = \frac{5 \times 1}{-4 \times 2} = \frac{5}{-8} = \frac{-5}{8} \][/tex]
Calculate [tex]\(a \times (b \times c)\)[/tex]:
[tex]\[ a \times (b \times c) = \frac{-7}{3} \times \frac{-5}{8} = \frac{-7 \times -5}{3 \times 8} = \frac{35}{24} \approx 1.4583333333333335 \][/tex]
Since both sides are identical:
[tex]\[ a \times (b \times c) = a \times (b \times c) = \frac{35}{24} \][/tex]
### Property (c): [tex]\(a \times (b + c) = (a \times b) + (a \times c)\)[/tex]
1. Calculate [tex]\(b + c\)[/tex] again:
[tex]\[ b + c = \frac{5}{-4} + \frac{1}{2} = \frac{-3}{4} \][/tex]
2. Calculate [tex]\(a \times (b + c)\)[/tex]:
[tex]\[ a \times (b + c) = \frac{-7}{3} \times \frac{-3}{4} = \frac{-7 \times -3}{3 \times 4} = \frac{21}{12} = \frac{7}{4} = 1.75 \][/tex]
3. Calculate [tex]\(a \times b\)[/tex]:
[tex]\[ a \times b = \frac{-7}{3} \times \frac{5}{-4} = \frac{-7 \times 5}{3 \times -4} = \frac{-35}{-12} = \frac{35}{12} \approx 2.9166666666666665 \][/tex]
4. Calculate [tex]\(a \times c\)[/tex]:
[tex]\[ a \times c = \frac{-7}{3} \times \frac{1}{2} = \frac{-7 \times 1}{3 \times 2} = \frac{-7}{6} \approx -1.1666666666666667 \][/tex]
5. Calculate [tex]\((a \times b) + (a \times c)\)[/tex]:
[tex]\[ (a \times b) + (a \times c) = \frac{35}{12} + \frac{-7}{6} = \frac{35}{12} + \frac{-14}{12} = \frac{21}{12} = 1.7500000000000002 \][/tex]
The discrepancy arises due to rounding errors in decimal approximations but they are conceptually equal:
[tex]\[ a \times (b + c) = 1.75 \quad \text{and} \quad (a \times b) + (a \times c) = 1.7500000000000002 \][/tex]
Thus although minor rounding differences might appear, we can conclude:
[tex]\[ a \times (b + c) \approx (a \times b) + (a \times c) \][/tex]
To summarize, we have verified that:
- (a) [tex]\(a + (b + c) = (a + b) + c\)[/tex] holds true.
- (b) [tex]\(a \times (b \times c) = a \times (b \times c)\)[/tex] holds true.
- (c) [tex]\(a \times (b + c) = (a \times b) + (a \times c)\)[/tex] is approximately true, acknowledging minor numerical approximation differences.
### Property (a): [tex]\(a + (b + c) = (a + b) + c\)[/tex]
We need to check if the associative property of addition holds.
1. Calculate [tex]\(b + c\)[/tex]:
[tex]\[ b + c = \frac{5}{-4} + \frac{1}{2} = \frac{-5}{4} + \frac{1}{2} = \frac{-5}{4} + \frac{2}{4} = \frac{-3}{4} \][/tex]
2. Calculate [tex]\(a + (b + c)\)[/tex]:
[tex]\[ a + (b + c) = \frac{-7}{3} + \frac{-3}{4} = \frac{-28}{12} + \frac{-9}{12} = \frac{-37}{12} \approx -3.0833333333333335 \][/tex]
3. Calculate [tex]\(a + b\)[/tex]:
[tex]\[ a + b = \frac{-7}{3} + \frac{5}{-4} = \frac{-28}{12} + \frac{-15}{12} = \frac{-43}{12} \][/tex]
4. Calculate [tex]\((a + b) + c\)[/tex]:
[tex]\[ (a + b) + c = \frac{-43}{12} + \frac{1}{2} = \frac{-43}{12} + \frac{6}{12} = \frac{-37}{12} \approx -3.0833333333333335 \][/tex]
Therefore:
[tex]\[ a + (b + c) = \frac{-37}{12} \quad \text{and} \quad (a + b) + c = \frac{-37}{12} \][/tex]
Since both sides are equal:
[tex]\[ a + (b + c) = (a + b) + c \][/tex]
### Property (b): [tex]\(a \times (b \times c) = a \times (b \times c)\)[/tex]
This property seems straightforward as it involves the same exact terms on both sides.
Calculate [tex]\(b \times c\)[/tex]:
[tex]\[ b \times c = \frac{5}{-4} \times \frac{1}{2} = \frac{5 \times 1}{-4 \times 2} = \frac{5}{-8} = \frac{-5}{8} \][/tex]
Calculate [tex]\(a \times (b \times c)\)[/tex]:
[tex]\[ a \times (b \times c) = \frac{-7}{3} \times \frac{-5}{8} = \frac{-7 \times -5}{3 \times 8} = \frac{35}{24} \approx 1.4583333333333335 \][/tex]
Since both sides are identical:
[tex]\[ a \times (b \times c) = a \times (b \times c) = \frac{35}{24} \][/tex]
### Property (c): [tex]\(a \times (b + c) = (a \times b) + (a \times c)\)[/tex]
1. Calculate [tex]\(b + c\)[/tex] again:
[tex]\[ b + c = \frac{5}{-4} + \frac{1}{2} = \frac{-3}{4} \][/tex]
2. Calculate [tex]\(a \times (b + c)\)[/tex]:
[tex]\[ a \times (b + c) = \frac{-7}{3} \times \frac{-3}{4} = \frac{-7 \times -3}{3 \times 4} = \frac{21}{12} = \frac{7}{4} = 1.75 \][/tex]
3. Calculate [tex]\(a \times b\)[/tex]:
[tex]\[ a \times b = \frac{-7}{3} \times \frac{5}{-4} = \frac{-7 \times 5}{3 \times -4} = \frac{-35}{-12} = \frac{35}{12} \approx 2.9166666666666665 \][/tex]
4. Calculate [tex]\(a \times c\)[/tex]:
[tex]\[ a \times c = \frac{-7}{3} \times \frac{1}{2} = \frac{-7 \times 1}{3 \times 2} = \frac{-7}{6} \approx -1.1666666666666667 \][/tex]
5. Calculate [tex]\((a \times b) + (a \times c)\)[/tex]:
[tex]\[ (a \times b) + (a \times c) = \frac{35}{12} + \frac{-7}{6} = \frac{35}{12} + \frac{-14}{12} = \frac{21}{12} = 1.7500000000000002 \][/tex]
The discrepancy arises due to rounding errors in decimal approximations but they are conceptually equal:
[tex]\[ a \times (b + c) = 1.75 \quad \text{and} \quad (a \times b) + (a \times c) = 1.7500000000000002 \][/tex]
Thus although minor rounding differences might appear, we can conclude:
[tex]\[ a \times (b + c) \approx (a \times b) + (a \times c) \][/tex]
To summarize, we have verified that:
- (a) [tex]\(a + (b + c) = (a + b) + c\)[/tex] holds true.
- (b) [tex]\(a \times (b \times c) = a \times (b \times c)\)[/tex] holds true.
- (c) [tex]\(a \times (b + c) = (a \times b) + (a \times c)\)[/tex] is approximately true, acknowledging minor numerical approximation differences.
