Join the conversation on IDNLearn.com and get the answers you seek from experts. Join our platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
Sure, let's prove the distributive property of multiplication over addition for the numbers [tex]\(2\)[/tex], [tex]\(\frac{4}{15}\)[/tex], and [tex]\(\frac{7}{15}\)[/tex].
The distributive property of multiplication over addition states that for any three real numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex],
[tex]\[ a \cdot (b + c) = (a \cdot b) + (a \cdot c) \][/tex]
Let's assign the values as follows:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = \frac{4}{15} \][/tex]
[tex]\[ c = \frac{7}{15} \][/tex]
We need to verify that:
[tex]\[ 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) = \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) \][/tex]
First, let's calculate the left side of the equation:
### Left Side: [tex]\( a \cdot (b + c) \)[/tex]
[tex]\[ a \cdot (b + c) = 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) \][/tex]
Combine the fractions inside the parentheses:
[tex]\[ \frac{4}{15} + \frac{7}{15} = \frac{4 + 7}{15} = \frac{11}{15} \][/tex]
Now multiply by [tex]\( a = 2 \)[/tex]:
[tex]\[ 2 \cdot \frac{11}{15} = \frac{2 \cdot 11}{15} = \frac{22}{15} \approx 1.4666666666666668 \][/tex]
### Right Side: [tex]\((a \cdot b) + (a \cdot c)\)[/tex]
Calculate each term individually:
[tex]\[ a \cdot b = 2 \cdot \frac{4}{15} = \frac{2 \cdot 4}{15} = \frac{8}{15} \approx 0.5333333333333333 \][/tex]
[tex]\[ a \cdot c = 2 \cdot \frac{7}{15} = \frac{2 \cdot 7}{15} = \frac{14}{15} \approx 0.9333333333333333 \][/tex]
Add these results together:
[tex]\[ \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) = \frac{8}{15} + \frac{14}{15} = \frac{8 + 14}{15} = \frac{22}{15} \approx 1.4666666666666668 \][/tex]
### Conclusion
Both the left side and the right side of the equation simplify to [tex]\(\frac{22}{15}\)[/tex], and their approximate decimal values are [tex]\(1.4666666666666668\)[/tex].
Therefore, the distributive property holds true in this case:
[tex]\[ 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) = \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) \][/tex]
So, we have proven that the distributive property of multiplication over addition is valid for the given numbers.
The distributive property of multiplication over addition states that for any three real numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex],
[tex]\[ a \cdot (b + c) = (a \cdot b) + (a \cdot c) \][/tex]
Let's assign the values as follows:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = \frac{4}{15} \][/tex]
[tex]\[ c = \frac{7}{15} \][/tex]
We need to verify that:
[tex]\[ 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) = \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) \][/tex]
First, let's calculate the left side of the equation:
### Left Side: [tex]\( a \cdot (b + c) \)[/tex]
[tex]\[ a \cdot (b + c) = 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) \][/tex]
Combine the fractions inside the parentheses:
[tex]\[ \frac{4}{15} + \frac{7}{15} = \frac{4 + 7}{15} = \frac{11}{15} \][/tex]
Now multiply by [tex]\( a = 2 \)[/tex]:
[tex]\[ 2 \cdot \frac{11}{15} = \frac{2 \cdot 11}{15} = \frac{22}{15} \approx 1.4666666666666668 \][/tex]
### Right Side: [tex]\((a \cdot b) + (a \cdot c)\)[/tex]
Calculate each term individually:
[tex]\[ a \cdot b = 2 \cdot \frac{4}{15} = \frac{2 \cdot 4}{15} = \frac{8}{15} \approx 0.5333333333333333 \][/tex]
[tex]\[ a \cdot c = 2 \cdot \frac{7}{15} = \frac{2 \cdot 7}{15} = \frac{14}{15} \approx 0.9333333333333333 \][/tex]
Add these results together:
[tex]\[ \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) = \frac{8}{15} + \frac{14}{15} = \frac{8 + 14}{15} = \frac{22}{15} \approx 1.4666666666666668 \][/tex]
### Conclusion
Both the left side and the right side of the equation simplify to [tex]\(\frac{22}{15}\)[/tex], and their approximate decimal values are [tex]\(1.4666666666666668\)[/tex].
Therefore, the distributive property holds true in this case:
[tex]\[ 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) = \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) \][/tex]
So, we have proven that the distributive property of multiplication over addition is valid for the given numbers.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
