Join the IDNLearn.com community and start finding the answers you need today. Discover detailed answers to your questions with our extensive database of expert knowledge.
Sagot :
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.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
